8.1 Simplify Expressions with Roots

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,

$\begin{matrix} 13^{2} is read “13 squared” \\ 169 is called the square of 13, since 13^{2} = 169 \\ 13 is a square root of 169 \end{matrix}$

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. This symbol, as well as other radicals to be introduced later, are grouping symbols.

We also use the radical sign for the square root of zero. Because $0 squared equals 0,$ $\sqrt{0} = 0 .$ Notice that zero has only one 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, $− \sqrt{169} = −13 .$

Can we simplify $the square root of −49 ?$ Is there a number whose square is $−49 ?$

$open parenthesis close parenthesis squared equals −49$

Any positive number squared is positive. Any negative number squared is positive. There is no real number equal to $the square root of −49.$ The square root of a negative number is not a real number.

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

Let’s review some vocabulary first.

$\begin{matrix} We write: & We say: \\ n^{2} & n squared \\ n^{3} & n cubed \\ n^{4} & n to the fourth power \\ n^{5} & n to the fifth power \end{matrix}$

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 Figure 8.2.

The figure contains two tables. The first table has 9 rows and 5 columns. The first row is a header row with the headers “Number”, “Square”, “Cube”, “Fourth power”, and “Fifth power”. The second row contains the expressions n, n squared, n cubed, n to the fourth power, and n to the fifth power. The third row contains the number 1 in each column. The fourth row contains the numbers 2, 4, 8, 16, 32. The fifth row contains the numbers 3, 9, 27, 81, 243. The sixth row contains the numbers 4, 16, 64, 256, 1024. The seventh row contains the numbers 5, 25, 125 625, 3125. The eighth row contains the expressions x, x squared, x cubed, x to the fourth power, and x to the fifth power. The last row contains the expressions x squared, x to the fourth power, x to the sixth power, x to the eighth power, and x to the tenth power. The second table has 7 rows and 5 columns. The first row is a header row with the headers “Number”, “Square”, “Cube”, “Fourth power”, and “Fifth power”. The second row contains the expressions n, n squared, n cubed, n to the fourth power, and n to the fifth power. The third row contains the numbers negative 1, 1 negative 1, 1, negative 1. The fourth row contains the numbers negative 2, 4, negative 8, 16, negative 32. The fifth row contains the numbers negative 3, 9, negative 27, 81, negative 243. The sixth row contains the numbers negative 4, 16, negative 64, 256, negative 1024. The last row contains the numbers negative 5, 25, negative 125, 625, negative 3125. — **Figure 8.2**

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.

The image contains a table with 2 rows and 5 columns. The first row contains the expressions n, n squared, n cubed, n to the fourth power, and n to the fifth power. The second row contains the numbers negative 2, 4, negative 8, 16, negative 32. Arrows point to the second and fourth columns with the label “Even power Positive result”. Arrows point to the first third and fifth columns with the label “Odd power Negative result”.

We will now extend the square root definition to higher roots.

Just like we use the word ‘cubed’ for b³, we use the term ‘cube root’ for $\sqrt[3]{a} .$

We can refer to Figure 8.2 to help find higher roots.

$\begin{matrix} 4^{3} & = & 64 \\ 3^{4} & = & 81 \\ {(−2)}^{5} & = & −32 \end{matrix} \begin{matrix} \sqrt[3]{64} & = & 4 \\ \sqrt[4]{81} & = & 3 \\ \sqrt[5]{−32} & = & −2 \end{matrix}$

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.

We will apply these properties in the next two examples.

In this example be alert for the negative signs as well as even and odd powers.

Example 4

Simplify: ⓐ $the cube root of −125$ ⓑ $\sqrt[4]{- 16}$ ⓒ $the 5th root of −243.$

ⓐ

	$the cube root of −125$
Since $open parenthesis −5 close parenthesis cubed equals −125.$	$−5$

ⓑ

	$the 4th root of −16$
Think, ${(?)}^{4} = −16 .$ No real number raised to the fourth power is negative.	Not a real number.

ⓒ

	$the 5th root of −243$
Since ${(−3)}^{5} = −243 .$	$−3$

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 $\sqrt{4} = 2,$ what is the value of $the root of 21$ or $the cube root of 50 ?$ 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 $the square root of 11,$ 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.

The figure contains two tables. The first table has 5 rows and 2 columns. The first row is a header row with the headers “Number” and “Square Root”. The second row has the numbers 4 and 2. The third row is 9 and 3. The fourth row is 16 and 4. The last row is 25 and 5. A callout containing the number 11 is directed between the 9 and 16 in the first column. Another callout containing the number square root of 11 is directed between the 3 and 4 of the second column. Below the table are the inequalities 9 is less than 11 is less than 16 and 3 is less than square root of 11 is less than 4. The second table has 5 rows and 2 columns. The first row is a header row with the headers “Number” and “Cube Root”. The second row has the numbers 8 and 2. The third row is 27 and 3. The fourth row is 64 and 4. The last row is 125 and 5. A callout containing the number 91 is directed between the 64 and 125 in the first column. Another callout containing the number cube root of 91 is directed between the 4 and 5 of the second column. Below the table are the inequalities 64 is less than 91 is less than 125 and 4 is less than cube root of 91 is less than 5.

Similarly, to estimate $the cube root of 91,$ we see 91 is between perfect cube numbers 64 and 125. The cube root then will be between 4 and 5.

Example 5

Estimate each root between two consecutive whole numbers: ⓐ $the square root of 105$ ⓑ $the cube root of 43.$

ⓐ Think of the perfect square numbers closest to 105. Make a small table of these perfect squares and their squares roots.



Locate 105 between two consecutive perfect squares.
$the square root of 105$ is between their square roots.

ⓑ Similarly we locate 43 between two perfect cube numbers.



Locate 43 between two consecutive perfect cubes.
$the cube root of 43$ is between their cube roots.

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 $\sqrt[y]{x}$ key.

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 $\approx$ and it is read ‘approximately’.

Suppose your calculator has a 10 digit display. You would see that

$\begin{matrix} \sqrt{5} \approx 2.236067978 rounded to two decimal places is \sqrt{5} \approx 2.24 \\ \sqrt[4]{93} \approx 3.105422799 rounded to two decimal places is \sqrt[4]{93} \approx 3.11 \end{matrix}$

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

$\begin{matrix} {(2.236067978)}^{2} & = & 5.000000002 \\ {(2.24)}^{2} & = & 5.0176 \end{matrix} \begin{matrix} {(3.105422799)}^{4} & = & 92.999999991 \\ {(3.11)}^{4} & = & 93.54951841 \end{matrix}$

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.

Example 6

Round to two decimal places: ⓐ $the square root of 17$ ⓑ $the cube root of 49$ ⓒ $the 4th root of 51.$

ⓐ

	$the square root of 17$
Use the calculator square root key.	$4.123105626 …$
Round to two decimal places.	$4.12$
	$\sqrt{17} \approx 4.12$

ⓑ

	$the cube root of 49$
Use the calculator $\sqrt[y]{x}$ key.	$3.659305710 …$
Round to two decimal places.	$3.66$
	$\sqrt[3]{49} \approx 3.66$

ⓒ

	$the 4th root of 51$
Use the calculator $\sqrt[y]{x}$ key.	$2.6723451177 …$
Round to two decimal places.	$2.67$
	$\sqrt[4]{51} \approx 2.67$

Simplify Variable Expressions with Roots

The odd root of a number can be either positive or negative. For example,

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 $\sqrt[4]{625} = 5 .$

But notice,

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. $| −5 | equals 5.$ So we say that when n is even $\sqrt[n]{a^{n}} = | a | .$ This guarantees the principal root is positive.

Example 7

Simplify: ⓐ $\sqrt{x^{2}}$ ⓑ $\sqrt[3]{n^{3}}$ ⓒ $\sqrt[4]{p^{4}}$ ⓓ $\sqrt[5]{y^{5}} .$

ⓐ We use the absolute value to be sure to get the positive root.

	$\sqrt{x^{2}}$
Since the index $n$ is even, $\sqrt[n]{a^{n}} = \| a \| .$	$\| x \|$

ⓑ This is an odd indexed root so there is no need for an absolute value sign.

	$\sqrt[3]{n^{3}}$
Since the index $n$ is odd, $\sqrt[n]{a^{n}} = a .$	$n$

ⓒ

	$\sqrt[4]{p^{4}}$
Since the index $n is even \sqrt[n]{a^{n}} = \| a \| .$	$\| p \|$

ⓓ

	$\sqrt[5]{y^{5}}$
Since the index $n$ is odd, $\sqrt[n]{a^{n}} = a .$	$y$

What about square roots of higher powers of variables? The Power Property of Exponents says ${(a^{m})}^{n} = a^{m \cdot n} .$ So if we square a^m, the exponent will become 2m.

${(a^{m})}^{2} = a^{2 m}$

Looking now at the square root,

$\begin{matrix} \sqrt{a^{2 m}} \\ Since {(a^{m})}^{2} = a^{2 m} . & \sqrt{{(a^{m})}^{2}} \\ Since n is even \sqrt[n]{a^{n}} = | a | . & | a^{m} | \\ So \sqrt{a^{2 m}} = | a^{m} | . \end{matrix}$

We apply this concept in the next example.

Example 8

Simplify: ⓐ $\sqrt{x^{6}}$ ⓑ $\sqrt{y^{16}} .$

ⓐ

	$\sqrt{x^{6}}$
Since ${(x^{3})}^{2} = x^{6} .$	$\sqrt{{(x^{3})}^{2}}$
Since the index $n$ is even $\sqrt{a^{n}} = \| a \| .$	$\| x cubed \|$

ⓑ

	$\sqrt{y^{16}}$
Since ${(y^{8})}^{2} = y^{16} .$	$\sqrt{{(y^{8})}^{2}}$
Since the index $n$ is even $\sqrt[n]{a^{n}} = \| a \| .$	$y to the power 8$
In this case the absolute value sign is not needed as $y to the power 8$ is positive.

The next example uses the same idea for higher roots.

In the next example, we now have a coefficient in front of the variable. The concept $\sqrt{a^{2 m}} = | a^{m} |$ works in much the same way.

$\sqrt{16 r^{22}} = 4 | r^{11} | because {(4 r^{11})}^{2} = 16 r^{22} .$

But notice $\sqrt{25 u^{8}} = 5 u^{4}$ and no absolute value sign is needed as u⁴ is always positive.

This example just takes the idea farther as it has roots of higher index.

The next examples have two variables.

Key Concepts

Square Root Notation
- $\sqrt{m}$ is read ‘the square root of m’
- If n² = m, then $n = \sqrt{m},$ for $n greater than or equal to 0.$
- The square root of m, $\sqrt{m},$ is a positive number whose square is m.
n^th Root of a Number
- If $b^{n} = a,$ then b is an n^th root of a.
- The principal n^th root of a is written $\sqrt[n]{a} .$
- n is called the index of the radical.
Properties of $\sqrt[n]{a}$
- When n is an even number and
  - $a greater than or equal to 0,$ then $\sqrt[n]{a}$ is a real number
  - $a less than 0,$ then $\sqrt[n]{a}$ is not a real number
- When n is an odd number, $\sqrt[n]{a}$ is a real number for all values of a.
Simplifying Odd and Even Roots
- For any integer n≥2,
  - when n is odd $\sqrt[n]{a^{n}} = a$
  - when n is even $\sqrt[n]{a^{n}} = | a |$
- We must use the absolute value signs when we take an even root of an expression with a variable in the radical.

Practice Makes Perfect

Simplify Expressions with Roots

In the following exercises, simplify.

ⓐ $the square root of 64$ ⓑ $− the square root of 81$

ⓐ 8 ⓑ $−9$

ⓐ $the square root of 169$ ⓑ $− the square root of 100$

ⓐ $the square root of 196$ ⓑ $− the square root of 1$

ⓐ 14 ⓑ $−1$

ⓐ $the square root of 144$ ⓑ $− the square root of 121$

ⓐ $\sqrt{\frac{4}{9}}$ ⓑ $− the square root of 0.01$

ⓐ $\frac{2}{3}$ ⓑ $−0.1$

ⓐ $\sqrt{\frac{64}{121}}$ ⓑ $− the square root of 0.16$

ⓐ $the square root of −121$ ⓑ $− the square root of 289$

ⓐ not real number ⓑ $−17$

ⓐ $− the square root of 400$ ⓑ $the square root of −36$

ⓐ $− the square root of 225$ ⓑ $the square root of −9$

ⓐ $−15$ ⓑ not real number

ⓐ $the square root of −49$ ⓑ $− the square root of 256$

ⓐ $the cube root of 216$ ⓑ $the 4th root of 256$

ⓐ 6 ⓑ 4

ⓐ $the cube root of 27$ ⓑ $the 4th root of 16$ ⓒ $the 5th root of 243$

ⓐ $the cube root of 512$ ⓑ $the 4th root of 81$ ⓒ $the 5th root of 1$

ⓐ 8 ⓑ 3 ⓒ 1

ⓐ $the cube root of 125$ ⓑ $the 4th root of 1296$ ⓒ $the 5th root of 1024$

ⓐ $the cube root of −8$ ⓑ $the 4th root of −81$ ⓒ $the 5th root of −32$

ⓐ $−2$ ⓑ $not real$ ⓒ $−2$

ⓐ $the cube root of −64$ ⓑ $the 4th root of −16$ ⓒ $the 5th root of −243$

ⓐ $the cube root of −125$ ⓑ $the 4th root of −1296$ ⓒ $the 5th root of −1024$

ⓐ $−5$ ⓑ $not real$ ⓒ $−4$

ⓐ $the cube root of −512$ ⓑ $the 4th root of −81$ ⓒ $the 5th root of −1$

Estimate and Approximate Roots

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

ⓐ $the square root of 70$ ⓑ $the cube root of 71$

ⓐ $8 less than the square root of 70 less than 9$
ⓑ $4 less than the cube root of 71 less than 5$

ⓐ $the square root of 55$ ⓑ $the cube root of 119$

ⓐ $the square root of 200$ ⓑ $the cube root of 137$

ⓐ $14 less than the square root of 200 less than 15$
ⓑ $5 less than the cube root of 137 less than 6$

ⓐ $the square root of 172$ ⓑ $the cube root of 200$

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

Simplify Variable Expressions with Roots

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

ⓐ $\sqrt[5]{u^{5}}$ ⓑ $\sqrt[8]{v^{8}}$

ⓐ u ⓑ $| v |$

ⓐ $\sqrt[3]{a^{3}}$ ⓑ $\sqrt[9]{b^{9}}$

ⓐ $\sqrt[4]{y^{4}}$ ⓑ $\sqrt[7]{m^{7}}$

ⓐ $| y |$ ⓑ $m$

ⓐ $\sqrt[8]{k^{8}}$ ⓑ $\sqrt[6]{p^{6}}$

ⓐ $\sqrt{x^{6}}$ ⓑ $\sqrt{y^{16}}$

ⓐ $| x cubed |$ ⓑ $y to the power 8$

ⓐ $\sqrt{a^{14}}$ ⓑ $\sqrt{w^{24}}$

ⓐ $\sqrt{x^{24}}$ ⓑ $\sqrt{y^{22}}$

ⓐ $x to the power 12$ ⓑ $| y to the power 11 |$

ⓐ $\sqrt{a^{12}}$ ⓑ $\sqrt{b^{26}}$

ⓐ $\sqrt[3]{x^{9}}$ ⓑ $\sqrt[4]{y^{12}}$

ⓐ $x^{3}$ ⓑ $| y cubed |$

ⓐ $\sqrt[5]{a^{10}}$ ⓑ $\sqrt[3]{b^{27}}$

ⓐ $\sqrt[4]{m^{8}}$ ⓑ $\sqrt[5]{n^{20}}$

ⓐ $m^{2}$ ⓑ $n to the power 4$

ⓐ $\sqrt[6]{r^{12}}$ ⓑ $\sqrt[3]{s^{30}}$

ⓐ $the square root of 49 x squared$ ⓑ $− \sqrt{81 x^{18}}$

ⓐ $7 | x |$ ⓑ $−9 | x to the power 9 |$

ⓐ $the square root of 100 y squared$ ⓑ $− \sqrt{100 m^{32}}$

ⓐ $\sqrt{121 m^{20}}$ ⓑ $− the square root of 64 a squared$

ⓐ $11 m to the power 10$ ⓑ $−8 | a |$

ⓐ $\sqrt{81 x^{36}}$ ⓑ $− the square root of 25 x squared$

ⓐ $\sqrt[4]{16 x^{8}}$ ⓑ $\sqrt[6]{64 y^{12}}$

ⓐ $2 x squared$ ⓑ $2 y squared$

ⓐ $\sqrt[3]{−8 c^{9}}$ ⓑ $\sqrt[3]{125 d^{15}}$

ⓐ $\sqrt[3]{216 a^{6}}$ ⓑ $\sqrt[5]{32 b^{20}}$

ⓐ $6 a squared$ ⓑ $2 b to the power 4$

ⓐ $\sqrt[7]{128 r^{14}}$ ⓑ $\sqrt[4]{81 s^{24}}$

Writing Exercises

Why is there no real number equal to $the square root of −64 ?$

Answers will vary.

What is the difference between $9 squared$ and $the square root of 9 ?$

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.

This table has 4 rows and 4 columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is “Confidently”, the third is “With some help”, and the fourth is “No, I don’t get it”. Under the first column are the phrases “simplify expressions with roots.”, “estimate and approximate roots”, and “simplify variable expressions with roots”. The other columns are left blank so that the learner may indicate their mastery level for each topic.

ⓑ 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. Whom 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.

	$\sqrt[3]{y^{18}}$
Since ${(y^{6})}^{3} = y^{18} .$	$\sqrt[3]{{(y^{6})}^{3}}$
Since $n$ is odd, $\sqrt[n]{a^{n}} = a .$	$y to the power 6$

	$\sqrt[4]{z^{8}}$
Since ${(z^{2})}^{4} = z^{8} .$	$\sqrt[4]{{(z^{2})}^{4}}$
Since $z^{2}$ is positive, we do not need an absolute value sign.	$z^{2}$

	$the square root of 16 n squared$
Since $open parenthesis 4 n close parenthesis squared equals 16 n squared.$	$the square root of open parenthesis 4 n close parenthesis squared$
Since the index $n$ is even $\sqrt[n]{a^{n}} = \| a \| .$	$4 \| n \|$

	$- \sqrt{81 c^{2}}$
Since $open parenthesis 9 c close parenthesis squared equals 81 c squared.$	$- \sqrt{{(9 c)}^{2}}$
Since the index $n$ is even $\sqrt[n]{a^{n}} = \| a \| .$	$−9 \| c \|$

	$\sqrt[3]{64 p^{6}}$
Rewrite $64 p to the power 6$ as $open parenthesis 4 p squared close parenthesis cubed.$	$the cube root of open parenthesis 4 p squared close parenthesis cubed$
Take the cube root.	$4 p squared$

	$\sqrt[4]{16 q^{12}}$
Rewrite the radicand as a fourth power.	$\sqrt[4]{{(2 q^{3})}^{4}}$
Take the fourth root.	$2 \| q cubed \|$

	$the square root of 36 x squared y squared$
Since $open parenthesis 6 x y close parenthesis squared equals 36 x squared y squared$	$the square root of open parenthesis 6 x y close parenthesis squared$
Take the square root.	$6 \| x y \|$

	$\sqrt{121 a^{6} b^{8}}$
Since ${(11 a^{3} b^{4})}^{2} = 121 a^{6} b^{8}$	$\sqrt{{(11 a^{3} b^{4})}^{2}}$
Take the square root.	$11 \| a cubed \| b to the power 4$

	$\sqrt[3]{64 p^{63} q^{9}}$
Since ${(4 p^{21} q^{3})}^{3} = 64 p^{63} q^{9}$	$\sqrt[3]{{(4 p^{21} q^{3})}^{3}}$
Take the cube root.	$4 p to the power 21 q cubed$