Use the Complex Number System

By the end of this section, you will be able to:

Evaluate the square root of a negative number
Add and subtract complex numbers
Multiply complex numbers
Divide complex numbers
Simplify powers of $i$

Before you get started, take this readiness quiz.

Given the numbers $−4, - \sqrt{7}, 0. \bar{5}, \frac{7}{3}, 3, \sqrt{81},$
list the ⓐ rational numbers, ⓑ irrational numbers, ⓒ real numbers.

If you missed this problem, review [link].
Multiply: $(x - 3) (2 x + 5) .$

If you missed this problem, review [link].
Rationalize the denominator: $\frac{\sqrt{5}}{\sqrt{5} - \sqrt{3}} .$

If you missed this problem, review [link].

Evaluate the Square Root of a Negative Number

Whenever we have a situation where we have a square root of a negative number we say there is no real number that equals that square root. For example, to simplify $\sqrt{−1},$

we are looking for a real number x so that x² = –1. Since all real numbers squared are positive numbers, there is no real number that equals –1 when squared.

Mathematicians have often expanded their numbers systems as needed. They added 0 to the counting numbers to get the whole numbers. When they needed negative balances, they added negative numbers to get the integers. When they needed the idea of parts of a whole they added fractions and got the rational numbers. Adding the irrational numbers allowed numbers like $\sqrt{5} .$

All of these together gave us the real numbers and so far in your study of mathematics, that has been sufficient.

But now we will expand the real numbers to include the square roots of negative numbers. We start by defining the imaginary unit $i$

as the number whose square is –1.

Imaginary Unit

The imaginary unit i is the number whose square is –1.

i^{2} = −1 or i = \sqrt{−1}

We will use the imaginary unit to simplify the square roots of negative numbers.

Square Root of a Negative Number

If b is a positive real number, then

\sqrt{− b} = \sqrt{b} i

We will use this definition in the next example. Be careful that it is clear that the i is not under the radical. Sometimes you will see this written as $\sqrt{− b} = i \sqrt{b}$

to emphasize the i is not under the radical. But the $\sqrt{− b} = \sqrt{b} i$

is considered standard form.

Write each expression in terms of i and simplify if possible:

ⓐ $\sqrt{−25}$

ⓑ $\sqrt{−7}$

ⓐ* * *

\begin{matrix} \sqrt{−25} \\ \begin{matrix} Use the definition of the square root of \\ negative numbers. \end{matrix} & \sqrt{25} i \\ Simplify. & 5 i \end{matrix}

ⓑ* * *

\begin{matrix} \sqrt{−7} \\ \begin{matrix} Use the definition of the square root of \\ negative numbers. \end{matrix} & \sqrt{7} i \\ Simplify. & \begin{matrix} Be careful that it is clear that i is not under the \\ radical sign. \end{matrix} \end{matrix}

ⓒ* * *

\begin{matrix} \sqrt{−12} \\ \begin{matrix} Use the definition of the square root of \\ negative numbers. \end{matrix} & \sqrt{12} i \\ Simplify \sqrt{12} . & 2 \sqrt{3} i \end{matrix}

Write each expression in terms of i and simplify if possible:

ⓐ $\sqrt{−81}$

ⓑ $\sqrt{−5}$

ⓐ $9 i$

ⓑ $\sqrt{5} i$

Write each expression in terms of i and simplify if possible:

ⓐ $\sqrt{−36}$

ⓑ $\sqrt{−3}$

ⓐ $6 i$

ⓑ $\sqrt{3} i$

Now that we are familiar with the imaginary number i, we can expand the real numbers to include imaginary numbers. The complex number system includes the real numbers and the imaginary numbers. A complex number is of the form a + bi, where a, b are real numbers. We call a the real part and b the imaginary part.

Complex Number

A complex number is of the form a + bi, where a and b are real numbers.

The image shows the expression a plus b i. The number a is labeled “real part” and the number b i is labeled “imaginary part”.

A complex number is in standard form when written as $a + b i,$

where a and b are real numbers.

If $b = 0,$

then $a + b i$

becomes $a + 0 \cdot i = a,$

and is a real number.

If $b \neq 0,$

then $a + b i$

is an imaginary number.

If $a = 0,$

then $a + b i$

becomes $0 + b i = b i,$

and is called a pure imaginary number.

We summarize this here.

a + b i


{: valign=”top”}	$b = 0$

\begin{matrix} a + 0 \cdot i \\ a \end{matrix}

Real number
{: valign=”top”}	$b \neq 0$

a + b i

Imaginary number
{: valign=”top”}	$a = 0$

\begin{array}{l} 0 + b i \\ b i \end{array}

| Pure imaginary number | {: valign=”top”}{: .unnumbered summary=”The table has four rows and three columns. The first row is a header and the second column entry a plus b i. In the second row is b equals zero, a plus 0 i, and “Real number”. The third row contains b is not equal to 0, a plus b i, and “Imaginary number”. The fourth row contains a = 0, 0 plus b i, and “Pure imaginary number”.”}

The standard form of a complex number is $a + b i,$

so this explains why the preferred form is $\sqrt{− b} = \sqrt{b} i$

when $b > 0 .$

The diagram helps us visualize the complex number system. It is made up of both the real numbers and the imaginary numbers.

The table has four rows and three columns. The first row is a header and the second column entry a plus b i. In the second row is b equals zero, a plus 0 i, and “Real number”. The third row contains b is not equal to 0, a plus b i, and “Imaginary number”. The fourth row contains a = 0, 0 plus b i, and “Pure imaginary number”. ### Add or Subtract Complex Numbers

We are now ready to perform the operations of addition, subtraction, multiplication and division on the complex numbers—just as we did with the real numbers.

Adding and subtracting complex numbers is much like adding or subtracting like terms. We add or subtract the real parts and then add or subtract the imaginary parts. Our final result should be in standard form.

Add: $\sqrt{−12} + \sqrt{−27} .$

\begin{matrix} \sqrt{−12} + \sqrt{−27} \\ \begin{matrix} Use the definition of the square root of \\ negative numbers. \end{matrix} & \sqrt{12} i + \sqrt{27} i \\ Simplify the square roots. & 2 \sqrt{3} i + 3 \sqrt{3} i \\ Add. & 5 \sqrt{3} i \end{matrix}

Add: $\sqrt{−8} + \sqrt{−32} .$

6 \sqrt{2} i

Add: $\sqrt{−27} + \sqrt{−48} .$

7 \sqrt{3} i

Remember to add both the real parts and the imaginary parts in this next example.

Simplify: ⓐ $(4 - 3 i) + (5 + 6 i)$

ⓑ $(2 - 5 i) - (5 - 2 i) .$

ⓐ* * *

\begin{matrix} (4 - 3 i) + (5 + 6 i) \\ \begin{matrix} Use the Associative Property to put the real \\ parts and the imaginary parts together. \end{matrix} & (4 + 5) + (−3 i + 6 i) \\ Simplify. & 9 + 3 i \end{matrix}

ⓑ* * *

\begin{matrix} (2 - 5 i) - (5 - 2 i) \\ Distribute. & 2 - 5 i - 5 + 2 i \\ \begin{matrix} Use the Associative Property to put the real \\ parts and the imaginary parts together. \end{matrix} & 2 - 5 - 5 i + 2 i \\ Simplify. & −3 - 3 i \end{matrix}

Simplify: ⓐ $(2 + 7 i) + (4 - 2 i)$

ⓑ $(8 - 4 i) - (2 - i) .$

ⓐ $6 + 5 i$

ⓑ $6 - 3 i$

Simplify: ⓐ $(3 - 2 i) + (−5 - 4 i)$

ⓑ $(4 + 3 i) - (2 - 6 i) .$

ⓐ $−2 - 6 i$

ⓑ $2 + 9 i$

Multiply Complex Numbers

Multiplying complex numbers is also much like multiplying expressions with coefficients and variables. There is only one special case we need to consider. We will look at that after we practice in the next two examples.

Multiply: $2 i (7 - 5 i) .$

\begin{matrix} 2 i (7 - 5 i) \\ Distribute. & 14 i - 10 i^{2} \\ Simplify i^{2} . & 14 i - 10 (−1) \\ Multiply. & 14 i + 10 \\ Write in standard form. & 10 + 14 i \end{matrix}

Multiply: $4 i (5 - 3 i) .$

12 + 20 i

Multiply: $−3 i (2 + 4 i) .$

12 + 6 i

In the next example, we multiply the binomials using the Distributive Property or FOIL.

Multiply: $(3 + 2 i) (4 - 3 i) .$

\begin{matrix} (3 + 2 i) (4 - 3 i) \\ Use FOIL. & 12 - 9 i + 8 i - 6 i^{2} \\ Simplify i^{2} and combine like terms. & 12 - i - 6 (−1) \\ Multiply. & 12 - i + 6 \\ Combine the real parts. & 18 - i \end{matrix}

Multiply: $(5 - 3 i) (−1 - 2 i) .$

−11 - 7 i

Multiply: $(−4 - 3 i) (2 + i) .$

−5 - 10 i

In the next example, we could use FOIL or the Product of Binomial Squares Pattern.

Multiply: ${(3 + 2 i)}^{2}$


Use the Product of Binomial Squares Pattern, ${(a + b)}^{2} = a^{2} + 2 a b + b^{2} .$


Simplify.
Simplify $i^{2} .$


Simplify.

Multiply using the Binomial Squares pattern: ${(−2 - 5 i)}^{2} .$

−21 - 20 i

Multiply using the Binomial Squares pattern: ${(−5 + 4 i)}^{2} .$

9 - 40 i

Since the square root of a negative number is not a real number, we cannot use the Product Property for Radicals. In order to multiply square roots of negative numbers we should first write them as complex numbers, using $\sqrt{− b} = \sqrt{b} i .$

This is one place students tend to make errors, so be careful when you see multiplying with a negative square root.

Multiply: $\sqrt{−36} \cdot \sqrt{−4} .$

To multiply square roots of negative numbers, we first write them as complex numbers.

\begin{matrix} \sqrt{−36} \cdot \sqrt{−4} \\ Write as complex numbers using \sqrt{− b} = \sqrt{b} i . & \sqrt{36} i \cdot \sqrt{4} i \\ Simplify. & 6 i \cdot 2 i \\ Multiply. & 12 i^{2} \\ Simplify i^{2} and multiply. & −12 \end{matrix}

Multiply: $\sqrt{−49} \cdot \sqrt{−4} .$

−14

Multiply: $\sqrt{−36} \cdot \sqrt{−81} .$

−54

In the next example, each binomial has a square root of a negative number. Before multiplying, each square root of a negative number must be written as a complex number.

Multiply: $(3 - \sqrt{−12}) (5 + \sqrt{−27}) .$

To multiply square roots of negative numbers, we first write them as complex numbers.

\begin{matrix} (3 - \sqrt{−12}) (5 + \sqrt{−27}) \\ Write as complex numbers using \sqrt{− b} = \sqrt{b} i . & (3 - 2 \sqrt{3} i) (5 + 3 \sqrt{3} i) \\ Use FOIL. & 15 + 9 \sqrt{3} i - 10 \sqrt{3} i - 6 \cdot 3 i^{2} \\ Combine like terms and simplify i^{2} . & 15 - \sqrt{3} i - 6 \cdot (−3) \\ Multiply and combine like terms. & 33 - \sqrt{3} i \end{matrix}

Multiply: $(4 - \sqrt{−12}) (3 - \sqrt{−48}) .$

−12 - 22 \sqrt{3} i

Multiply: $(−2 + \sqrt{−8}) (3 - \sqrt{−18}) .$

6 + 12 \sqrt{2} i

We first looked at conjugate pairs when we studied polynomials. We said that a pair of binomials that each have the same first term and the same last term, but one is a sum and one is a difference is called a conjugate pair and is of the form $(a - b), (a + b) .$

A complex conjugate pair is very similar. For a complex number of the form $a + b i,$

its conjugate is $a - b i .$

Notice they have the same first term and the same last term, but one is a sum and one is a difference.

Complex Conjugate Pair

A complex conjugate pair is of the form $a + b i,$

a - b i .

We will multiply a complex conjugate pair in the next example.

Multiply: $(3 - 2 i) (3 + 2 i) .$

\begin{matrix} (3 - 2 i) (3 + 2 i) \\ Use FOIL. & 9 + 6 i - 6 i - 4 i^{2} \\ Combine like terms and simplify i^{2} . & 9 - 4 (−1) \\ Multiply and combine like terms. & 13 \end{matrix}

Multiply: $(4 - 3 i) \cdot (4 + 3 i) .$

Multiply: $(−2 + 5 i) \cdot (−2 - 5 i) .$

From our study of polynomials, we know the product of conjugates is always of the form $(a - b) (a + b) = a^{2} - b^{2} .$

The result is called a difference of squares. We can multiply a complex conjugate pair using this pattern.

The last example we used FOIL. Now we will use the Product of Conjugates Pattern.

Notice this is the same result we found in [link].

When we multiply complex conjugates, the product of the last terms will always have an $i^{2}$

which simplifies to $−1 .$

\begin{matrix} (a - b i) (a + b i) \\ a^{2} - {(b i)}^{2} \\ a^{2} - b^{2} i^{2} \\ a^{2} - b^{2} (−1) \\ a^{2} + b^{2} \end{matrix}

This leads us to the Product of Complex Conjugates Pattern: $(a - b i) (a + b i) = a^{2} + b^{2}$

Product of Complex Conjugates

If a and b are real numbers, then

(a - b i) (a + b i) = a^{2} + b^{2}

Multiply using the Product of Complex Conjugates Pattern: $(8 - 2 i) (8 + 2 i) .$


Use the Product of Complex Conjugates Pattern, $(a - b i) (a + b i) = a^{2} + b^{2} .$
Simplify the squares.
Add.

Multiply using the Product of Complex Conjugates Pattern: $(3 - 10 i) (3 + 10 i) .$

109

Multiply using the Product of Complex Conjugates Pattern: $(−5 + 4 i) (−5 - 4 i) .$

Divide Complex Numbers

Dividing complex numbers is much like rationalizing a denominator. We want our result to be in standard form with no imaginary numbers in the denominator.

How to Divide Complex Numbers

Divide: $\frac{4 + 3 i}{3 - 4 i} .$

![Step 1 is to write both the numerator and denominator in standard form. For this example they are both in standard form.](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_08_08_006a_img.jpg) ![Step 2 is to multiply the numerator and denominator by the complex conjugate of the denominator. The complex conjugate of 3 minus 4 i is 3 plus 4 i. The resulting expression is the quantity 4 plus 3 i in parentheses times the quantity 3 plus 4 i in parentheses divided by the product of 3 minus 4 i in parentheses and the quantity 3 plus 4 i in parentheses.](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_08_08_006b_img.jpg) ![Step 3 is to simplify and write the result in standard form. Use the pattern the quantity a plus b i in parentheses equals a squared plus b squared in the denominator. The expression for this example then becomes the quantity 12 plus 16 i plus 9 i plus 12 i squared in parentheses divided by the sum of 9 and 16. Combining like terms we get the quantity 12 plus 25 i minus 12 in parentheses divided by 25. Simplifying we get 25 i divided by 25. Write the result in standard form. The result is i.](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_08_08_006c_img.jpg)

Divide: $\frac{2 + 5 i}{5 - 2 i} .$

Divide: $\frac{1 + 6 i}{6 - i} .$

We summarize the steps here.

How to divide complex numbers.

Write both the numerator and denominator in standard form.
Multiply the numerator and denominator by the complex conjugate of the denominator.
Simplify and write the result in standard form.

Divide, writing the answer in standard form: $\frac{−3}{5 + 2 i} .$

\begin{matrix} \frac{−3}{5 + 2 i} \\ \begin{matrix} Multiply the numerator and denominator by the \\ complex conjugate of the denominator. \end{matrix} & \frac{−3 (5 - 2 i)}{(5 + 2 i) (5 - 2 i)} \\ \begin{matrix} Multiply in the numerator and use the Product of \\ Complex Conjugates Pattern in the denominator. \end{matrix} & \frac{−15 + 6 i}{5^{2} + 2^{2}} \\ Simplify. & \frac{−15 + 6 i}{29} \\ Write in standard form. & - \frac{15}{29} + \frac{6}{29} i \end{matrix}

Divide, writing the answer in standard form: $\frac{4}{1 - 4 i} .$

\frac{4}{17} + \frac{16}{17} i

Divide, writing the answer in standard form: $\frac{−2}{−1 + 2 i} .$

\frac{2}{5} + \frac{4}{5} i

Be careful as you find the conjugate of the denominator.

Divide: $\frac{5 + 3 i}{4 i} .$

\begin{matrix} \frac{5 + 3 i}{4 i} \\ Write the denominator in standard form. & \frac{5 + 3 i}{0 + 4 i} \\ \begin{matrix} Multiply the numerator and denominator by \\ the complex conjugate of the denominator. \end{matrix} & \frac{(5 + 3 i) (0 - 4 i)}{(0 + 4 i) (0 - 4 i)} \\ Simplify. & \frac{(5 + 3 i) (−4 i)}{(4 i) (−4 i)} \\ Multiply. & \frac{−20 i - 12 i^{2}}{−16 i^{2}} \\ Simplify the i^{2} . & \frac{−20 i + 12}{16} \\ Rewrite in standard form. & \frac{12}{16} - \frac{20}{16} i \\ Simplify the fractions. & \frac{3}{4} - \frac{5}{4} i \end{matrix}

Divide: $\frac{3 + 3 i}{2 i} .$

\frac{3}{2} - \frac{3}{2} i

Divide: $\frac{2 + 4 i}{5 i} .$

\frac{4}{5} - \frac{2}{5} i

Simplify Powers of i

The powers of $i$

make an interesting pattern that will help us simplify higher powers of i. Let’s evaluate the powers of $i$

to see the pattern.

\begin{matrix} i^{1} & i^{2} & i^{3} & i^{4} \\ i & - 1 & i^{2} \cdot i & i^{2} \cdot i^{2} \\ - 1 \cdot i & (−1) (−1) \\ - i & 1 \\ i^{5} & i^{6} & i^{7} & i^{8} \\ i^{4} \cdot i & i^{4} \cdot i^{2} & i^{4} \cdot i^{3} & i^{4} \cdot i^{4} \\ 1 \cdot i & 1 \cdot i^{2} & 1 \cdot i^{3} & 1 \cdot 1 \\ i & i^{2} & i^{3} & 1 \\ - 1 & - i \end{matrix}

We summarize this now.

\begin{matrix} i^{1} & = & i & i^{5} & = & i \\ i^{2} & = & −1 & i^{6} & = & −1 \\ i^{3} & = & − i & i^{7} & = & − i \\ i^{4} & = & 1 & i^{8} & = & 1 \end{matrix}

If we continued, the pattern would keep repeating in blocks of four. We can use this pattern to help us simplify powers of i. Since i⁴ = 1, we rewrite each power, iⁿ, as a product using i⁴ to a power and another power of i.

We rewrite it in the form $i^{n} = {(i^{4})}^{q} \cdot i^{r},$

where the exponent, q, is the quotient of n divided by 4 and the exponent, r, is the remainder from this division. For example, to simplify i⁵⁷, we divide 57 by 4 and we get 14 with a remainder of 1. In other words, $57 = 4 \cdot 14 + 1 .$

So we write $i^{57} = {(1^{4})}^{14} \cdot i^{1}$

and then simplify from there.

Simplify: $i^{86} .$

\begin{matrix} i^{86} \\ \begin{matrix} Divide 86 by 4 and rewrite i^{86} in the \\ i^{n} = {(i^{4})}^{q} \cdot i^{r} form. \end{matrix} & {(1^{4})}^{21} \cdot i^{2} \end{matrix}

\begin{matrix} Simplify. & {(1)}^{21} \cdot (−1) \\ Simplify. & –1 \end{matrix}

</div>

Simplify: $i^{75} .$

− i

Simplify: $i^{92} .$

1

Access these online resources for additional instruction and practice with the complex number system.

Key Concepts

Square Root of a Negative Number
- If b is a positive real number, then $\sqrt{− b} = \sqrt{b} i$
$a + b i$

{: valign=”top”} $b = 0$

$\begin{matrix} a + 0 \cdot i \\ a \end{matrix}$

Real number

{: valign=”top”} $b \neq 0$

$a + b i$

Imaginary number

{: valign=”top”} $a = 0$

$\begin{array}{l} 0 + b i \\ b i \end{array}$

| Pure imaginary number | {: valign=”top”}{: summary=”The table has four rows and three columns. The first row is a header and the second column entry a plus b i. In the second row is b equals zero, a plus 0 i, and “Real number”. The third row contains b is not equal to 0, a plus b i, and “Imaginary number”. The fourth row contains a = 0, 0 plus b i, and “Pure imaginary number”.”}
- A complex number is in standard form when written as a + bi, where a, b are real numbers.
Product of Complex Conjugates
- If a, b are real numbers, then
  $(a - b i) (a + b i) = a^{2} + b^{2}$
How to Divide Complex Numbers
1. Write both the numerator and denominator in standard form.
2. Multiply the numerator and denominator by the complex conjugate of the denominator.
3. Simplify and write the result in standard form.

Section Exercises

Practice Makes Perfect

Evaluate the Square Root of a Negative Number

In the following exercises, write each expression in terms of i and simplify if possible.

ⓐ $\sqrt{−16}$

ⓑ $\sqrt{−11}$

ⓐ $4 i$

ⓑ $\sqrt{11} i$

ⓐ $\sqrt{−121}$

ⓑ $\sqrt{−1}$

ⓐ $\sqrt{−100}$

ⓑ $\sqrt{−13}$

ⓐ $10 i$

ⓑ $\sqrt{13} i$

ⓐ $\sqrt{−49}$

ⓑ $\sqrt{−15}$

Add or Subtract Complex Numbers In the following exercises, add or subtract.

\sqrt{−75} + \sqrt{−48}

9 \sqrt{3} i

\sqrt{−12} + \sqrt{−75}

\sqrt{−50} + \sqrt{−18}

8 \sqrt{2} i

\sqrt{−72} + \sqrt{−8}

(1 + 3 i) + (7 + 4 i)

8 + 7 i

(6 + 2 i) + (3 - 4 i)

(8 - i) + (6 + 3 i)

14 + 2 i

(7 - 4 i) + (−2 - 6 i)

(1 - 4 i) - (3 - 6 i)

−2 + 2 i

(8 - 4 i) - (3 + 7 i)

(6 + i) - (−2 - 4 i)

8 + 5 i

(−2 + 5 i) - (−5 + 6 i)

(5 - \sqrt{−36}) + (2 - \sqrt{−49})

7 - 13 i

(−3 + \sqrt{−64}) + (5 - \sqrt{−16})

(−7 - \sqrt{−50}) - (−32 - \sqrt{−18})

25 - 2 \sqrt{2} i

(−5 + \sqrt{−27}) - (−4 - \sqrt{−48})

Multiply Complex Numbers

In the following exercises, multiply.

4 i (5 - 3 i)

12 + 20 i

2 i (−3 + 4 i)

−6 i (−3 - 2 i)

−12 + 18 i

− i (6 + 5 i)

(4 + 3 i) (−5 + 6 i)

−38 + + 9 i

(−2 - 5 i) (−4 + 3 i)

(−3 + 3 i) (−2 - 7 i)

27 + 15 i

(−6 - 2 i) (−3 - 5 i)

In the following exercises, multiply using the Product of Binomial Squares Pattern.

{(3 + 4 i)}^{2}

−7 + 24 i

{(−1 + 5 i)}^{2}

{(−2 - 3 i)}^{2}

−5 - 12 i

{(−6 - 5 i)}^{2}

In the following exercises, multiply.

\sqrt{−25} \cdot \sqrt{−36}

−11

\sqrt{−4} \cdot \sqrt{−16}

\sqrt{−9} \cdot \sqrt{−100}

−30

\sqrt{−64} \cdot \sqrt{−9}

(−2 - \sqrt{−27}) (4 - \sqrt{−48})

−44 + 4 \sqrt{3} i

(5 - \sqrt{−12}) (−3 + \sqrt{−75})

(2 + \sqrt{−8}) (−4 + \sqrt{−18})

−20 - 2 \sqrt{2} i

(5 + \sqrt{−18}) (−2 - \sqrt{−50})

(2 - i) (2 + i)

(4 - 5 i) (4 + 5 i)

(7 - 2 i) (7 + 2 i)

(−3 - 8 i) (−3 + 8 i)

In the following exercises, multiply using the Product of Complex Conjugates Pattern.

(7 - i) (7 + i)

(6 - 5 i) (6 + 5 i)

(9 - 2 i) (9 + 2 i)

(−3 - 4 i) (−3 + 4 i)

Divide Complex Numbers

In the following exercises, divide.

\frac{3 + 4 i}{4 - 3 i}

\frac{5 - 2 i}{2 + 5 i}

\frac{2 + i}{3 - 4 i}

\frac{2}{25} + \frac{11}{25} i

\frac{3 - 2 i}{6 + i}

\frac{3}{2 - 3 i}

\frac{6}{13} + \frac{9}{13} i

\frac{2}{4 - 5 i}

\frac{−4}{3 - 2 i}

- \frac{12}{13} - \frac{8}{13} i

\frac{−1}{3 + 2 i}

\frac{1 + 4 i}{3 i}

\frac{4}{3} - \frac{1}{3} i

\frac{4 + 3 i}{7 i}

\frac{−2 - 3 i}{4 i}

- \frac{3}{4} + \frac{1}{2} i

\frac{−3 - 5 i}{2 i}

Simplify Powers of i

In the following exercises, simplify.

i^{41}

i^{39}

i^{66}

−1

i^{48}

i^{128}

i^{162}

i^{137}

i^{255}

Writing Exercises

Explain the relationship between real numbers and complex numbers.

Answers will vary.

Aniket multiplied as follows and he got the wrong answer. What is wrong with his reasoning?

\begin{matrix} \sqrt{−7} \cdot \sqrt{−7} \\ \sqrt{49} \\ 7 \end{matrix}

Why is $\sqrt{−64} = 8 i$

but $\sqrt[3]{−64} = −4 .$

Answers will vary.

Explain how dividing complex numbers is similar to rationalizing a denominator.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ On a scale of $1 - 10,$

how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

Chapter Review Exercises

Simplify Expressions with Roots

Simplify Expressions with Roots

In the following exercises, simplify.

ⓐ $\sqrt{225}$

ⓑ $− \sqrt{16}$

ⓐ 15 ⓑ $−4$

ⓐ $− \sqrt{169}$

ⓑ $\sqrt{−8}$

ⓐ $\sqrt[3]{8}$

ⓑ $\sqrt[4]{81}$

ⓐ 2 ⓑ 3 ⓒ 3

ⓐ $\sqrt[3]{−512}$

ⓑ $\sqrt[4]{−81}$

Estimate and Approximate Roots

In the following exercises, estimate each root between two consecutive whole numbers.

ⓐ $\sqrt{68}$

ⓑ $\sqrt[3]{84}$

ⓐ $8 < \sqrt{68} < 9$

ⓑ $4 < \sqrt[3]{84} < 5$

In the following exercises, approximate each root and round to two decimal places.

ⓐ $\sqrt{37}$

ⓑ $\sqrt[3]{84}$

Simplify Variable Expressions with Roots

In the following exercises, simplify using absolute values as necessary.

ⓐ $\sqrt[3]{a^{3}}$

ⓑ $\sqrt[7]{b^{7}}$

ⓐ a ⓑ $\| b \|$

ⓐ $\sqrt{a^{14}}$

ⓑ $\sqrt{w^{24}}$

ⓐ $\sqrt[4]{m^{8}}$

ⓑ $\sqrt[5]{n^{20}}$

ⓐ $m^{2}$

ⓑ $n^{4}$

ⓐ $\sqrt{121 m^{20}}$

ⓑ $− \sqrt{64 a^{2}}$

ⓐ $\sqrt[3]{216 a^{6}}$

ⓑ $\sqrt[5]{32 b^{20}}$

ⓐ $6 a^{2}$

ⓑ $2 b^{4}$

ⓐ $\sqrt{144 x^{2} y^{2}}$

ⓑ $\sqrt{169 w^{8} y^{10}}$

Simplify Radical Expressions

Use the Product Property to Simplify Radical Expressions

In the following exercises, use the Product Property to simplify radical expressions.

\sqrt{125}

5 \sqrt{5}

\sqrt{675}

ⓐ $\sqrt[3]{625}$

ⓑ $\sqrt[6]{128}$

ⓐ $5 \sqrt[3]{5}$

ⓑ $2 \sqrt[6]{2}$

In the following exercises, simplify using absolute value signs as needed.

ⓐ $\sqrt{a^{23}}$

ⓑ $\sqrt[3]{b^{8}}$

ⓐ $\sqrt{80 s^{15}}$

ⓑ $\sqrt[5]{96 a^{7}}$

ⓐ $4 \| s^{7} \| \sqrt[]{5 s}$

ⓑ $2 a \sqrt[5]{3 a^{2}}$

ⓐ $\sqrt{96 r^{3} s^{3}}$

ⓑ $\sqrt[3]{80 x^{7} y^{6}}$

ⓐ $\sqrt[5]{−32}$

ⓑ $\sqrt[8]{−1}$

ⓐ $−2$

ⓑ not real

ⓐ $8 + \sqrt{96}$

ⓑ $\frac{2 + \sqrt{40}}{2}$

Use the Quotient Property to Simplify Radical Expressions

In the following exercises, use the Quotient Property to simplify square roots.

ⓐ $\sqrt{\frac{72}{98}}$

ⓑ $\sqrt[3]{\frac{24}{81}}$

ⓐ $\frac{6}{7}$

ⓑ $\frac{2}{3}$

ⓐ $\sqrt{\frac{y^{4}}{y^{8}}}$

ⓑ $\sqrt[5]{\frac{u^{21}}{u^{11}}}$

\sqrt{\frac{300 m^{5}}{64}}

\frac{10 m^{2} \sqrt{3 m}}{8}

ⓐ $\sqrt{\frac{28 p^{7}}{q^{2}}}$

ⓑ $\sqrt[3]{\frac{81 s^{8}}{t^{3}}}$

ⓐ $\sqrt{\frac{27 p^{2} q}{108 p^{4} q^{3}}}$

ⓑ $\sqrt[3]{\frac{16 c^{5} d^{7}}{250 c^{2} d^{2}}}$

ⓐ $\frac{1}{2 \| p q \|}$

ⓑ $\frac{2 c d \sqrt[5]{2 d^{2}}}{5}$

ⓐ $\frac{\sqrt{80 q^{5}}}{\sqrt{5 q}}$

ⓑ $\frac{\sqrt[3]{- 625}}{\sqrt[3]{5}}$

Simplify Rational Exponents

**Simplify expressions with $a^{\frac{1}{n}}$

In the following exercises, write as a radical expression.

ⓐ $r^{\frac{1}{2}}$

ⓑ $s^{\frac{1}{3}}$

ⓐ $\sqrt{r}$

ⓑ $\sqrt[3]{s}$

In the following exercises, write with a rational exponent.

ⓐ $\sqrt{21 p}$

ⓑ $\sqrt[4]{8 q}$

In the following exercises, simplify.

ⓐ $625^{\frac{1}{4}}$

ⓑ $243^{\frac{1}{5}}$

ⓐ 5 ⓑ 3 ⓒ 2

ⓐ ${(−1,000)}^{\frac{1}{3}}$

ⓑ $− {1,000}^{\frac{1}{3}}$

ⓐ ${(−32)}^{\frac{1}{5}}$

ⓑ ${(243)}^{- \frac{1}{5}}$

ⓐ $−2$

ⓑ $\frac{1}{3}$

**Simplify Expressions with $a^{\frac{m}{n}}$

In the following exercises, write with a rational exponent.

ⓐ $\sqrt[4]{r^{7}}$

ⓑ ${(\sqrt[5]{2 p q})}^{3}$

In the following exercises, simplify.

ⓐ $25^{\frac{3}{2}}$

ⓑ $9^{- \frac{3}{2}}$

ⓐ 125 ⓑ $\frac{1}{27}$

ⓐ $− 64^{\frac{3}{2}}$

ⓑ $− 64^{- \frac{3}{2}}$

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

ⓐ $6^{\frac{5}{2}} \cdot 6^{\frac{1}{2}}$

ⓑ ${(b^{15})}^{\frac{3}{5}}$

ⓐ $6^{3}$

ⓑ $b^{9}$

ⓐ $\frac{a^{\frac{3}{4}} \cdot a^{- \frac{1}{4}}}{a^{- \frac{10}{4}}}$

ⓑ ${(\frac{27 b^{\frac{2}{3}} c^{- \frac{5}{2}}}{b^{- \frac{7}{3}} c^{\frac{1}{2}}})}^{\frac{1}{3}}$

Add, Subtract and Multiply Radical Expressions

Add and Subtract Radical Expressions

In the following exercises, simplify.

ⓐ $7 \sqrt{2} - 3 \sqrt{2}$

ⓑ $7 \sqrt[3]{p} + 2 \sqrt[3]{p}$

ⓐ $4 \sqrt{2}$

ⓑ $9 \sqrt[3]{p}$

ⓐ $\sqrt{11 b} - 5 \sqrt{11 b} + 3 \sqrt{11 b}$

ⓑ $8 \sqrt[4]{11 c d} + 5 \sqrt[4]{11 c d} - 9 \sqrt[4]{11 c d}$

ⓐ $\sqrt{48} + \sqrt{27}$

ⓑ $\sqrt[3]{54} + \sqrt[3]{128}$

ⓐ $7 \sqrt{3}$

ⓑ $7 \sqrt[3]{2}$

ⓐ $\sqrt{80 c^{7}} - \sqrt{20 c^{7}}$

ⓑ $2 \sqrt[4]{162 r^{10}} + 4 \sqrt[4]{32 r^{10}}$

3 \sqrt{75 y^{2}} + 8 y \sqrt{48} - \sqrt{300 y^{2}}

37 y \sqrt{3}

Multiply Radical Expressions

In the following exercises, simplify.

ⓐ $(5 \sqrt{6}) (− \sqrt{12})$

ⓑ $(−2 \sqrt[4]{18}) (− \sqrt[4]{9})$

ⓐ $(3 \sqrt{2 x^{3}}) (7 \sqrt{18 x^{2}})$

ⓑ $(−6 \sqrt[3]{20 a^{2}}) (−2 \sqrt[3]{16 a^{3}})$

ⓐ $126 x^{2} \sqrt{2}$

ⓑ $48 a \sqrt[3]{a^{2}}$

Use Polynomial Multiplication to Multiply Radical Expressions

In the following exercises, multiply.

ⓐ $\sqrt{11} (8 + 4 \sqrt{11})$

ⓑ $\sqrt[3]{3} (\sqrt[3]{9} + \sqrt[3]{18})$

ⓐ $(3 - 2 \sqrt{7}) (5 - 4 \sqrt{7})$

ⓑ $(\sqrt[3]{x} - 5) (\sqrt[3]{x} - 3)$

ⓐ $71 - 22 \sqrt{7}$

ⓑ $\sqrt[3]{x^{2}} - 8 \sqrt[3]{x} + 15$

(2 \sqrt{7} - 5 \sqrt{11}) (4 \sqrt{7} + 9 \sqrt{11})

ⓐ ${(4 + \sqrt{11})}^{2}$

ⓑ ${(3 - 2 \sqrt{5})}^{2}$

ⓐ $27 + 8 \sqrt{11}$

ⓑ $29 - 12 \sqrt{5}$

(7 + \sqrt{10}) (7 - \sqrt{10})

(\sqrt[3]{3 x} + 2) (\sqrt[3]{3 x} - 2)

\sqrt[3]{9 x^{2}} - 4

Divide Radical Expressions

Divide Square Roots

In the following exercises, simplify.

ⓐ $\frac{\sqrt{48}}{\sqrt{75}}$

ⓑ $\frac{\sqrt[3]{81}}{\sqrt[3]{24}}$

ⓐ $\frac{\sqrt{320 m n^{−5}}}{\sqrt{45 m^{−7} n^{3}}}$

ⓑ $\frac{\sqrt[3]{16 x^{4} y^{−2}}}{\sqrt[3]{−54 x^{−2} y^{4}}}$

ⓐ $\frac{8 m^{4}}{3 n^{4}}$

ⓑ $- \frac{x^{2}}{2 y^{2}}$

Rationalize a One Term Denominator

In the following exercises, rationalize the denominator.

ⓐ $\frac{8}{\sqrt{3}}$

ⓑ $\sqrt{\frac{7}{40}}$

ⓐ $\frac{1}{\sqrt[3]{11}}$

ⓑ $\sqrt[3]{\frac{7}{54}}$

ⓐ $\frac{\sqrt[3]{121}}{11}$

ⓑ $\frac{\sqrt[3]{28}}{6}$

ⓐ $\frac{1}{\sqrt[4]{4}}$

ⓑ $\sqrt[4]{\frac{9}{32}}$

Rationalize a Two Term Denominator

In the following exercises, simplify.

\frac{7}{2 - \sqrt{6}}

- \frac{7 (2 + \sqrt{6})}{2}

\frac{\sqrt{5}}{\sqrt{n} - \sqrt{7}}

\frac{\sqrt{x} + \sqrt{8}}{\sqrt{x} - \sqrt{8}}

{\frac{(\sqrt{x} + 2 \sqrt{2})}{x - 8}}^{2}

Solve Radical Equations

Solve Radical Equations

In the following exercises, solve.

\sqrt{4 x - 3} = 7

\sqrt{5 x + 1} = −3

no solution

\sqrt[3]{4 x - 1} = 3

\sqrt{u - 3} + 3 = u

u = 3, u = 4

\sqrt[3]{4 x + 5} - 2 = −5

{(8 x + 5)}^{\frac{1}{3}} + 2 = −1

x = −4

\sqrt{y + 4} - y + 2 = 0

2 \sqrt{8 r + 1} - 8 = 2

r = 3

Solve Radical Equations with Two Radicals

In the following exercises, solve.

\sqrt{10 + 2 c} = \sqrt{4 c + 16}

\sqrt[3]{2 x^{2} + 9 x - 18} = \sqrt[3]{x^{2} + 3 x - 2}

x = −8, x = 2

\sqrt{r} + 6 = \sqrt{r + 8}

\sqrt{x + 1} - \sqrt{x - 2} = 1

x = 3

Use Radicals in Applications

In the following exercises, solve. Round approximations to one decimal place.

Landscaping Reed wants to have a square garden plot in his backyard. He has enough compost to cover an area of 75 square feet. Use the formula $s = \sqrt{A}$

to find the length of each side of his garden. Round your answer to the nearest tenth of a foot.

Accident investigation An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was 175 feet. Use the formula $s = \sqrt{24 d}$

to find the speed of the vehicle before the brakes were applied. Round your answer to the nearest tenth.

64.8

feet

Use Radicals in Functions

Evaluate a Radical Function

In the following exercises, evaluate each function.

g (x) = \sqrt{6 x + 1},

find* * *

ⓐ $g (4)$

ⓑ $g (8)$

G (x) = \sqrt{5 x - 1},

find* * *

ⓐ $G (5)$

ⓑ $G (2)$

ⓐ $G (5) = 2 \sqrt{6}$

ⓑ $G (2) = 3$

h (x) = \sqrt[3]{x^{2} - 4},

find* * *

ⓐ $h (−2)$

ⓑ $h (6)$

For the function* * *

g (x) = \sqrt[4]{4 - 4 x},

find* * *

ⓐ $g (1)$

ⓑ $g (−3)$

ⓐ $g (1) = 0$

ⓑ $g (−3) = 2$

Find the Domain of a Radical Function

In the following exercises, find the domain of the function and write the domain in interval notation.

g (x) = \sqrt{2 - 3 x}

F (x) = \sqrt{\frac{x + 3}{x - 2}}

(2, \infty)

f (x) = \sqrt[3]{4 x^{2} - 16}

F (x) = \sqrt[4]{10 - 7 x}

[\frac{7}{10}, \infty)

Graph Radical Functions

In the following exercises, ⓐ find the domain of the function ⓑ graph the function ⓒ use the graph to determine the range.

g (x) = \sqrt{x + 4}

g (x) = 2 \sqrt{x}

ⓐ domain: $[0, \infty)$

ⓑ* * *

The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from 0 to 8. The y-axis runs from 0 to 8. The function has a starting point at (0, 0) and goes through the points (1, 2) and (4, 4).

f (x) = \sqrt[3]{x - 1}

f (x) = \sqrt[3]{x} + 3

ⓐ domain: $(− \infty, \infty)$

ⓑ* * *

The figure shows a cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis runs from negative 2 to 6. The function has a center point at (0, 3) and goes through the points (negative 1, 2) and (1, 4).

Use the Complex Number System

Evaluate the Square Root of a Negative Number

In the following exercises, write each expression in terms of i and simplify if possible.

ⓐ $\sqrt{−100}$

ⓑ $\sqrt{−13}$

Add or Subtract Complex Numbers

In the following exercises, add or subtract.

\sqrt{−50} + \sqrt{−18}

8 \sqrt{2} i

(8 - i) + (6 + 3 i)

(6 + i) - (−2 - 4 i)

8 + 5 i

(−7 - \sqrt{−50}) - (−32 - \sqrt{−18})

Multiply Complex Numbers

In the following exercises, multiply.

(−2 - 5 i) (−4 + 3 i)

23 + 14 i

−6 i (−3 - 2 i)

\sqrt{−4} \cdot \sqrt{−16}

−6

(5 - \sqrt{−12}) (−3 + \sqrt{−75})

In the following exercises, multiply using the Product of Binomial Squares Pattern.

{(−2 - 3 i)}^{2}

−5 - 12 i

In the following exercises, multiply using the Product of Complex Conjugates Pattern.

(9 - 2 i) (9 + 2 i)

Divide Complex Numbers

In the following exercises, divide.

\frac{2 + i}{3 - 4 i}

\frac{2}{25} + \frac{11}{25} i

\frac{−4}{3 - 2 i}

Simplify Powers of i

In the following exercises, simplify.

i^{48}

i^{255}

Practice Test

In the following exercises, simplify using absolute values as necessary.

\sqrt[3]{125 x^{9}}

5 x^{3}

\sqrt{169 x^{8} y^{6}}

\sqrt[3]{72 x^{8} y^{4}}

2 x^{2} y^{} \sqrt[3]{9 x^{2} y}

\sqrt{\frac{45 x^{3} y^{4}}{180 x^{5} y^{2}}}

In the following exercises, simplify. Assume all variables are positive.

ⓐ $216^{- \frac{1}{4}}$

ⓑ $− 49^{\frac{3}{2}}$

ⓐ $\frac{1}{4}$

ⓑ $−343$

\sqrt{−45}

\frac{x^{- \frac{1}{4}} \cdot x^{\frac{5}{4}}}{x^{- \frac{3}{4}}}

x^{\frac{7}{4}}

{(\frac{8 ​ x^{\frac{2}{3}} ​ y^{- \frac{5}{2}}}{x^{- \frac{7}{3}} y^{\frac{1}{2}}})}^{\frac{1}{3}}

\sqrt{48 x^{5}} - \sqrt{75 x^{5}}

− x^{2} \sqrt{3 x}

\sqrt{27 x^{2}} - 4 x \sqrt{12} + \sqrt{108 x^{2}}

2 \sqrt{12 x^{5}} \cdot 3 \sqrt{6 x^{3}}

36 x^{4} \sqrt{2}

\sqrt[3]{4} (\sqrt[3]{16} - \sqrt[3]{6})

(4 - 3 \sqrt{3}) (5 + 2 \sqrt{3})

2 - 7 \sqrt{3}

\frac{\sqrt[3]{128}}{\sqrt[3]{54}}

\frac{\sqrt{245 x y^{−4}}}{\sqrt{45 x^{−4} y^{3}}}

\frac{7 x^{5}}{3 y^{7}}

\frac{1}{\sqrt[3]{5}}

\frac{3}{2 + \sqrt{3}}

3 (2 - \sqrt{3})

\sqrt{−4} \cdot \sqrt{−9}

−4 i (−2 - 3 i)

−12 + 8 i

\frac{4 + i}{3 - 2 i}

i^{172}

− i

In the following exercises, solve.

\sqrt{2 x + 5} + 8 = 6

\sqrt{x + 5} + 1 = x

x = 4

\sqrt[3]{2 x^{2} - 6 x - 23} = \sqrt[3]{x^{2} - 3 x + 5}

In the following exercise, ⓐ find the domain of the function ⓑ graph the function ⓒ use the graph to determine the range.

g (x) = \sqrt{x + 2}

ⓐ domain: $[−2, \infty)$

ⓑ* * *

The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 2 to 6. The y-axis runs from 0 to 8. The function has a starting point at (negative 2, 0) and goes through the points (negative 1, 1) and (2, 2).

Glossary

complex conjugate pair: A complex conjugate pair is of the form a + bi, a – bi.

complex number: A complex number is of the form a + bi, where a and b are real numbers. We call a the real part and b the imaginary part.

complex number system: The complex number system is made up of both the real numbers and the imaginary numbers.

imaginary unit: The imaginary unit $i$
is the number whose square is –1. i² = –1 or
$i = \sqrt{−1} .$

standard form: A complex number is in standard form when written as $a + b i,$
where a, b are real numbers.

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