In this section students will:
A hardware store sells 16-ft ladders and 24-ft ladders. A window is located 12 feet above the ground. A ladder needs to be purchased that will reach the window from a point on the ground 5 feet from the building. To find out the length of ladder needed, we can draw a right triangle as shown in [link], and use the Pythagorean Theorem.
Now, we need to find out the length that, when squared, is 169, to determine which ladder to choose. In other words, we need to find a square root. In this section, we will investigate methods of finding solutions to problems such as this one.
When the square root of a number is squared, the result is the original number. Since
the square root of
is
The square root function is the inverse of the squaring function just as subtraction is the inverse of addition. To undo squaring, we take the square root.
In general terms, if
is a positive real number, then the square root of
is a number that, when multiplied by itself, gives
The square root could be positive or negative because multiplying two negative numbers gives a positive number. The principal square root is the nonnegative number that when multiplied by itself equals
The square root obtained using a calculator is the principal square root.
The principal square root of
is written as
The symbol is called a radical, the term under the symbol is called the radicand, and the entire expression is called a radical expression.
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The principal square root of
is the nonnegative number that, when multiplied by itself, equals
It is written as a radical expression, with a symbol called a radical over the term called the radicand:
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Does
No. Although both
and
are
the radical symbol implies only a nonnegative root, the principal square root. The principal square root of 25 is
Evaluate each expression.
because
because
and
because
because
and
**For
can we find the square roots before adding?**
No.
This is not equivalent to
The order of operations requires us to add the terms in the radicand before finding the square root.
Evaluate each expression.
To simplify a square root, we rewrite it such that there are no perfect squares in the radicand. There are several properties of square roots that allow us to simplify complicated radical expressions. The first rule we will look at is the product rule for simplifying square roots, which allows us to separate the square root of a product of two numbers into the product of two separate rational expressions. For instance, we can rewrite
as
We can also use the product rule to express the product of multiple radical expressions as a single radical expression.
If
and
are nonnegative, the square root of the product
is equal to the product of the square roots of
and
Given a square root radical expression, use the product rule to simplify it.
Simplify the radical expression.
Simplify
Notice the absolute value signs around x and y? That’s because their value must be positive!
Given the product of multiple radical expressions, use the product rule to combine them into one radical expression.
Simplify the radical expression.* * *
Simplify
assuming
Just as we can rewrite the square root of a product as a product of square roots, so too can we rewrite the square root of a quotient as a quotient of square roots, using the quotient rule for simplifying square roots. It can be helpful to separate the numerator and denominator of a fraction under a radical so that we can take their square roots separately. We can rewrite
as
The square root of the quotient
is equal to the quotient of the square roots of
and
where
Given a radical expression, use the quotient rule to simplify it.
Simplify the radical expression.
Simplify
We do not need the absolute value signs for
because that term will always be nonnegative.
Simplify the radical expression.
Simplify
We can add or subtract radical expressions only when they have the same radicand and when they have the same radical type such as square roots. For example, the sum of
and
is
However, it is often possible to simplify radical expressions, and that may change the radicand. The radical expression
can be written with a
in the radicand, as
so
Given a radical expression requiring addition or subtraction of square roots, solve.
Add
We can rewrite
as
According the product rule, this becomes
The square root of
is 2, so the expression becomes
which is
Now we can the terms have the same radicand so we can add.
Add
Subtract
Rewrite each term so they have equal radicands.
Now the terms have the same radicand so we can subtract.
Subtract
When an expression involving square root radicals is written in simplest form, it will not contain a radical in the denominator. We can remove radicals from the denominators of fractions using a process called rationalizing the denominator.
We know that multiplying by 1 does not change the value of an expression. We use this property of multiplication to change expressions that contain radicals in the denominator. To remove radicals from the denominators of fractions, multiply by the form of 1 that will eliminate the radical.
For a denominator containing a single term, multiply by the radical in the denominator over itself. In other words, if the denominator is
multiply by
For a denominator containing the sum or difference of a rational and an irrational term, multiply the numerator and denominator by the conjugate of the denominator, which is found by changing the sign of the radical portion of the denominator. If the denominator is
then the conjugate is
Given an expression with a single square root radical term in the denominator, rationalize the denominator.
Write
in simplest form.
The radical in the denominator is
So multiply the fraction by
Then simplify.
Write
in simplest form.
Given an expression with a radical term and a constant in the denominator, rationalize the denominator.
Write
in simplest form.
Begin by finding the conjugate of the denominator by writing the denominator and changing the sign. So the conjugate of
is
Then multiply the fraction by
Write
in simplest form.
Although square roots are the most common rational roots, we can also find cube roots, 4th roots, 5th roots, and more. Just as the square root function is the inverse of the squaring function, these roots are the inverse of their respective power functions. These functions can be useful when we need to determine the number that, when raised to a certain power, gives a certain number.
Suppose we know that
We want to find what number raised to the 3rd power is equal to 8. Since
we say that 2 is the cube root of 8.
The nth root of
is a number that, when raised to the nth power, gives
For example,
is the 5th root of
because
If
is a real number with at least one nth root, then the principal nth root of
is the number with the same sign as
that, when raised to the nth power, equals
The principal nth root of
is written as
where
is a positive integer greater than or equal to 2. In the radical expression,
is called the index of the radical.
If
is a real number with at least one nth root, then the principal nth root of
written as
is the number with the same sign as
that, when raised to the nth power, equals
The index of the radical is
Simplify each of the following:
because
because
Simplify.
Radical expressions can also be written without using the radical symbol. We can use rational (fractional) exponents. The index must be a positive integer. If the index
is even, then
cannot be negative.
We can also have rational exponents with numerators other than 1. In these cases, the exponent must be a fraction in lowest terms. We raise the base to a power and take an nth root. The numerator tells us the power and the denominator tells us the root.
All of the properties of exponents that we learned for integer exponents also hold for rational exponents.
Rational exponents are another way to express principal nth roots. The general form for converting between a radical expression with a radical symbol and one with a rational exponent is
Given an expression with a rational exponent, write the expression as a radical.
Write
as a radical. Simplify.
The 2 tells us the power and the 3 tells us the root.
We know that
because
Because the cube root is easy to find, it is easiest to find the cube root before squaring for this problem. In general, it is easier to find the root first and then raise it to a power.
Write
as a radical. Simplify.
Write
using a rational exponent.
The power is 2 and the root is 7, so the rational exponent will be
We get
Using properties of exponents, we get
Write
using a rational exponent.
Simplify:
Simplify
Access these online resources for additional instruction and practice with radicals and rational exponents.
is the nonnegative number that when multiplied by itself equals
See [link].
and
are nonnegative, the square root of the product
is equal to the product of the square roots of
and
and
are nonnegative, the square root of the quotient
is equal to the quotient of the square roots of
and
is the number with the same sign as
that when raised to the nth power equals
These roots have the same properties as square roots.**{::}** See [link].
What does it mean when a radical does not have an index? Is the expression equal to the radicand? Explain.
When there is no index, it is assumed to be 2 or the square root. The expression would only be equal to the radicand if the index were 1.
Where would radicals come in the order of operations? Explain why.
Every number will have two square roots. What is the principal square root?
The principal square root is the nonnegative root of the number.
Can a radical with a negative radicand have a real square root? Why or why not?
For the following exercises, simplify each expression.
16
10
14
25
For the following exercises, simplify each expression.
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So the length of the guy wire can be found by evaluating
What is the length of the guy wire?
500 feet
A car accelerates at a rate of
where t is the time in seconds after the car moves from rest. Simplify the expression.
For the following exercises, simplify each expression.
that when raised to the nth power equals
that, when multiplied by itself, equals
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