Simplify Expressions with Roots

In Foundations, we briefly looked at square roots. Remember that when a real number n is multiplied by itself, we write

n^{2}

and read it ‘n squared’. This number is called the square of n, and n is called the square root. For example,

Notice (−13)² = 169 also, so −13 is also a square root of 169. Therefore, both 13 and −13 are square roots of 169.

So, every positive number has two square roots—one positive and one negative. What if we only wanted the positive square root of a positive number? We use a radical sign, and write,

\sqrt{m},

which denotes the positive square root of m. The positive square root is also called the principal square root.

We know that every positive number has two square roots and the radical sign indicates the positive one. We write

\sqrt{169} = 13 .

If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example,

− \sqrt{169} = −13 .

Any positive number squared is positive. Any negative number squared is positive. There is no real number equal to

\sqrt{−49} .

So far we have only talked about squares and square roots. Let’s now extend our work to include higher powers and higher roots.

The terms ‘squared’ and ‘cubed’ come from the formulas for area of a square and volume of a cube.

It will be helpful to have a table of the powers of the integers from −5 to 5. See [link].

Notice the signs in the table. All powers of positive numbers are positive, of course. But when we have a negative number, the even powers are positive and the odd powers are negative. We’ll copy the row with the powers of −2 to help you see this.

Just like we use the word ‘cubed’ for b³, we use the term ‘cube root’ for

\sqrt[3]{a} .

Could we have an even root of a negative number? We know that the square root of a negative number is not a real number. The same is true for any even root. Even roots of negative numbers are not real numbers. Odd roots of negative numbers are real numbers.

Estimate and Approximate Roots

When we see a number with a radical sign, we often don’t think about its numerical value. While we probably know that the

\sqrt{4} = 2,

In some situations a quick estimate is meaningful and in others it is convenient to have a decimal approximation.

To get a numerical estimate of a square root, we look for perfect square numbers closest to the radicand. To find an estimate of

\sqrt{11},

we see 11 is between perfect square numbers 9 and 16, closer to 9. Its square root then will be between 3 and 4, but closer to 3.

we see 91 is between perfect cube numbers 64 and 125. The cube root then will be between 4 and 5.

There are mathematical methods to approximate square roots, but nowadays most people use a calculator to find square roots. To find a square root you will use the

\sqrt{x}

key on your calculator. To find a cube root, or any root with higher index, you will use the

\sqrt[y]{x}

When you use these keys, you get an approximate value. It is an approximation, accurate to the number of digits shown on your calculator’s display. The symbol for an approximation is

\approx

How do we know these values are approximations and not the exact values? Look at what happens when we square them:

Their squares are close to 5, but are not exactly equal to 5. The fourth powers are close to 93, but not equal to 93.

Simplify Variable Expressions with Roots

Three equivalent expressions are written: the cube root of 4 cubed, the cube root of 64, and 4. There are arrows pointing to the 4 that is cubed in the first expression and the 4 in the last expression labeling them as “same”. Three more equivalent expressions are also written: the cube root of the quantity negative 4 in parentheses cubed, the cube root of negative 64, and negative 4. The negative 4 in the first expression and the negative 4 in the last expression are labeled as being the “same”.

But what about an even root? We want the principal root, so

\sqrt[4]{625} = 5 .

Three equivalent expressions are written: the fourth root of the quantity 5 to the fourth power in parentheses, the fourth root of 625, and 5. There are arrows pointing to the 5 in the first expression and the 5 in the last expression labeling them as “same”. Three more equivalent expressions are also written: the fourth root of the quantity negative 5 in parentheses to the fourth power in parentheses, the fourth root of 625, and 5. The negative 5 in the first expression and the 5 in the last expression are labeled as being the “different”.

How can we make sure the fourth root of −5 raised to the fourth power is 5? We can use the absolute value.

\| −5 \| = 5 .

What about square roots of higher powers of variables? The Power Property of Exponents says

{(a^{m})}^{n} = a^{m \cdot n} .

In the next example, we now have a coefficient in front of the variable. The concept

\sqrt{a^{2 m}} = \| a^{m} \|

Key Concepts

Practice Makes Perfect

Simplify Expressions with Roots

In the following exercises, simplify.

ⓐ $\sqrt{64}$

ⓑ $− \sqrt{81}$

ⓐ 8 ⓑ $−9$

ⓐ $\sqrt{169}$

ⓑ $− \sqrt{100}$

ⓐ $\sqrt{196}$

ⓑ $− \sqrt{1}$

ⓐ 14 ⓑ $−1$

ⓐ $\sqrt{144}$

ⓑ $− \sqrt{121}$

ⓐ $\sqrt{\frac{4}{9}}$

ⓑ $− \sqrt{0.01}$

ⓐ $\frac{2}{3}$

ⓑ $−0.1$

ⓐ $\sqrt{\frac{64}{121}}$

ⓑ $− \sqrt{0.16}$

ⓐ $\sqrt{−121}$

ⓑ $− \sqrt{289}$

ⓐ not real number ⓑ $−17$

ⓐ $− \sqrt{400}$

ⓑ $\sqrt{−36}$

ⓐ $− \sqrt{225}$

ⓑ $\sqrt{−9}$

ⓐ $−15$

ⓑ not real number

ⓐ $\sqrt{−49}$

ⓑ $− \sqrt{256}$

ⓐ $\sqrt[3]{216}$

ⓑ $\sqrt[4]{256}$

ⓐ 6 ⓑ 4

ⓐ $\sqrt[3]{27}$

ⓑ $\sqrt[4]{16}$

ⓐ $\sqrt[3]{512}$

ⓑ $\sqrt[4]{81}$

ⓐ 8 ⓑ 3 ⓒ 1

ⓐ $\sqrt[3]{125}$

ⓑ $\sqrt[4]{1296}$

ⓐ $\sqrt[3]{−8}$

ⓑ $\sqrt[4]{−81}$

ⓐ $−2$

ⓑ $not real$

ⓐ $\sqrt[3]{−64}$

ⓑ $\sqrt[4]{−16}$

ⓐ $\sqrt[3]{−125}$

ⓑ $\sqrt[4]{−1296}$

ⓐ $−5$

ⓑ $not real$

ⓐ $\sqrt[3]{−512}$

ⓑ $\sqrt[4]{−81}$

Estimate and Approximate Roots

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

ⓐ $\sqrt{70}$

ⓑ $\sqrt[3]{71}$

ⓐ $8 < \sqrt{70} < 9$

ⓑ $4 < \sqrt[3]{71} < 5$

ⓐ $\sqrt{55}$

ⓑ $\sqrt[3]{119}$

ⓐ $\sqrt{200}$

ⓑ $\sqrt[3]{137}$

ⓐ $14 < \sqrt{200} < 15$

ⓑ $5 < \sqrt[3]{137} < 6$

ⓐ $\sqrt{172}$

ⓑ $\sqrt[3]{200}$

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

ⓐ $\sqrt{19}$

ⓑ $\sqrt[3]{89}$

ⓐ $4.36$

ⓑ $\approx 4.46$

ⓐ $\sqrt{21}$

ⓑ $\sqrt[3]{93}$

ⓐ $\sqrt{53}$

ⓑ $\sqrt[3]{147}$

ⓐ $7.28$

ⓑ $\approx 5.28$

ⓐ $\sqrt{47}$

ⓑ $\sqrt[3]{163}$

Simplify Variable Expressions with Roots

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

ⓐ $\sqrt[5]{u^{5}}$

ⓑ $\sqrt[8]{v^{8}}$

ⓐ u ⓑ $\| v \|$

ⓐ $\sqrt[3]{a^{3}}$

ⓑ $\sqrt[9]{b^{9}}$

ⓐ $\sqrt[4]{y^{4}}$

ⓑ $\sqrt[7]{m^{7}}$

ⓐ $\| y \|$

ⓑ $m$

ⓐ $\sqrt[8]{k^{8}}$

ⓑ $\sqrt[6]{p^{6}}$

ⓐ $\sqrt{x^{6}}$

ⓑ $\sqrt{y^{16}}$

ⓐ $\| x^{3} \|$

ⓑ $y^{8}$

ⓐ $\sqrt{a^{14}}$

ⓑ $\sqrt{w^{24}}$

ⓐ $\sqrt{x^{24}}$

ⓑ $\sqrt{y^{22}}$

ⓐ $x^{12}$

ⓑ $\| y^{11} \|$

ⓐ $\sqrt{a^{12}}$

ⓑ $\sqrt{b^{26}}$

ⓐ $\sqrt[3]{x^{9}}$

ⓑ $\sqrt[4]{y^{12}}$

ⓐ $x^{3}$

ⓑ $\| y^{3} \|$

ⓐ $\sqrt[5]{a^{10}}$

ⓑ $\sqrt[3]{b^{27}}$

ⓐ $\sqrt[4]{m^{8}}$

ⓑ $\sqrt[5]{n^{20}}$

ⓐ $m^{2}$

ⓑ $n^{4}$

ⓐ $\sqrt[6]{r^{12}}$

ⓑ $\sqrt[3]{s^{30}}$

ⓐ $\sqrt{49 x^{2}}$

ⓑ $− \sqrt{81 x^{18}}$

ⓐ $7 \| x \|$

ⓑ $−9 \| x^{9} \|$

ⓐ $\sqrt{100 y^{2}}$

ⓑ $− \sqrt{100 m^{32}}$

ⓐ $\sqrt{121 m^{20}}$

ⓑ $− \sqrt{64 a^{2}}$

ⓐ $11 m^{10}$

ⓑ $−8 \| a \|$

ⓐ $\sqrt{81 x^{36}}$

ⓑ $− \sqrt{25 x^{2}}$

ⓐ $\sqrt[4]{16 x^{8}}$

ⓑ $\sqrt[6]{64 y^{12}}$

ⓐ $2 x^{2}$

ⓑ $2 y^{2}$

ⓐ $\sqrt[3]{−8 c^{9}}$

ⓑ $\sqrt[3]{125 d^{15}}$

ⓐ $\sqrt[3]{216 a^{6}}$

ⓑ $\sqrt[5]{32 b^{20}}$

ⓐ $6 a^{2}$

ⓑ $2 b^{4}$

ⓐ $\sqrt[7]{128 r^{14}}$

ⓑ $\sqrt[4]{81 s^{24}}$

ⓐ $\sqrt{144 x^{2} y^{2}}$

ⓑ $\sqrt{169 w^{8} y^{10}}$

ⓐ $12 \| x y \|$

ⓑ $13 w^{4} \| y^{5} \|$

ⓐ $\sqrt{196 a^{2} b^{2}}$

ⓑ $\sqrt{81 p^{24} q^{6}}$

ⓐ $\sqrt{121 a^{2} b^{2}}$

ⓑ $\sqrt{9 c^{8} d^{12}}$

ⓐ $11 \| a b \|$

ⓑ $3 c^{4} d^{6}$

ⓐ $\sqrt{225 x^{2} y^{2} z^{2}}$

ⓑ $\sqrt{36 r^{6} s^{20}}$

Writing Exercises

Why is there no real number equal to $\sqrt{−64} ?$

Answers will vary.

What is the difference between $9^{2}$

and $\sqrt{9} ?$

Explain what is meant by the n^th root of a number.

Answers will vary.

Explain the difference of finding the n^th root of a number when the index is even compared to when the index is odd.

Self Check

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

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no - I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.


{: valign=”top”}
{: valign=”top”}	Locate 105 between two consecutive perfect squares.
{: valign=”top”}	$\sqrt{105}$


{: valign=”top”}
{: valign=”top”}	Locate 43 between two consecutive perfect cubes.
{: valign=”top”}	$\sqrt[3]{43}$