8.8 Use the Complex Number System

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 $the square root of −1,$ 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 $the square root of 5.$ 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.

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

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.

Example 1

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

ⓐ $the square root of −25$ ⓑ $the square root of −7$ ⓒ $the square root of −12.$

ⓐ

	$the square root of −25$
Use the definition of the square root of negative numbers.	$the square root of 25 i$
Simplify.	$5 i$

ⓑ

	$the square root of −7$
Use the definition of the square root of negative numbers.	$the square root of 7 i$
Simplify.	Be careful that it is clear that $i$ is not under the radical sign.

ⓒ

	$the square root of −12$
Use the definition of the square root of negative numbers.	$the square root of 12 i$
Simplify $the square root of 12.$	$2 the square root of 3 i$

Now that we are familiar with the imaginary number i, we can expand our concept of the number system 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.

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

If $b equals 0,$ then $a + b i$ becomes $a + 0 \cdot i = a,$ and is a real number.

If $b not equal to 0,$ then $a + b i$ is an imaginary number.

If $a equals 0,$ then $a + b i$ becomes $0 + b i = b i,$ and is called a pure imaginary number.

We summarize this here.

	$a + b i$
$b equals 0$	$\begin{matrix} a + 0 \cdot i \\ a \end{matrix}$	Real number
$b not equal to 0$	$a + b i$	Imaginary number
$a equals 0$	$0 plus b i; b i$	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 $b greater than 0.$

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.

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

Example 3

Simplify: ⓐ $(4 - 3 i) + (5 + 6 i)$ ⓑ $(2 - 5 i) - (5 - 2 i) .$

ⓐ

	$(4 - 3 i) + (5 + 6 i)$
Use the Associative Property to put the real parts and the imaginary parts together.	$(4 + 5) + (−3 i + 6 i)$
Simplify.	$9 plus 3 i$

ⓑ

	$(2 - 5 i) - (5 - 2 i)$
Distribute.	$2 - 5 i - 5 + 2 i$
Use the Associative Property to put the real parts and the imaginary parts together.	$2 - 5 - 5 i + 2 i$
Simplify.	$−3 minus 3 i$

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.

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

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

Since the square root of a negative number is not a real number, when we have the square roots of two negative numbers, 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.

Example 7

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

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

	$\sqrt{−36} \cdot \sqrt{−4}$
Write as complex numbers using $\sqrt{− b} = \sqrt{b} i .$	$\sqrt{36} i \cdot \sqrt{4} i$
Simplify.	$6 i times 2 i$
Multiply.	$12 i squared$
Simplify $i^{2}$ and multiply.	$−12$

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.

Example 8

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

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

	$(3 - \sqrt{−12}) (5 + \sqrt{−27})$
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.	$33 - \sqrt{3} i$

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.

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

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.

The quantity a minus b in parentheses times the quantity a plus b in parentheses is written above the expression showing the product of 3 minus 2 i in parentheses and 3 plus 2 i in parentheses. In the next line a squared minus b squared is written above the expression 3 squared minus the quantity 2 i in parentheses squared. Simplifying we get 9 minus 4 i squared. This is equal to 9 minus 4 times negative 1. The final result is 13.

Notice this is the same result we found in Example 9.

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

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.

We summarize the steps here.

Example 12

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

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

Be careful as you find the conjugate of the denominator.

Example 13

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

	$\frac{5 + 3 i}{4 i}$
Write the denominator in standard form.	$\frac{5 + 3 i}{0 + 4 i}$
Multiply the numerator and denominator by the complex conjugate of the denominator.	$\frac{(5 + 3 i) (0 - 4 i)}{(0 + 4 i) (0 - 4 i)}$
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.	$\frac{3}{4} - \frac{5}{4} i$

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.

A mathematical problem demonstrating the simplification of i to the power of 57, using long division (57 / 4 = 14 remainder 1) to determine that i^57 equals i.

Key Concepts

Square Root of a Negative Number
- If b is a positive real number, then $\sqrt{− b} = \sqrt{b} i$
  
  Table 8.1
  $a + b i$
  $b equals 0$ $\begin{matrix} a + 0 \cdot i \\ a \end{matrix}$ Real number
  $b not equal to 0$ $a + b i$ Imaginary number
  $a equals 0$ $0 plus b i; b i$ 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.

ⓐ $the square root of −16$ ⓑ $the square root of −11$
ⓒ $the square root of −8$

ⓐ $4 i$ ⓑ $the square root of 11 i$ ⓒ $2 the square root of 2 i$

ⓐ $the square root of −121$ ⓑ $the square root of −1$ ⓒ $the square root of −20$

ⓐ $the square root of −100$ ⓑ $the square root of −13$ ⓒ $the square root of −45$

ⓐ $10 i$ ⓑ $the square root of 13 i$ ⓒ $3 the square root of 5 i$

ⓐ $the square root of −49$ ⓑ $the square root of −15$ ⓒ $the square root of −75$

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

$\sqrt{−75} + \sqrt{−48}$

$9 the square root of 3 i$

$\sqrt{−12} + \sqrt{−75}$

$\sqrt{−50} + \sqrt{−18}$

$8 the square root of 2 i$

$\sqrt{−72} + \sqrt{−8}$

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

$8 plus 7 i$

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

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

$14 plus 2 i$

$(7 - 4 i) + (−2 - 6 i)$

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

$−2 plus 2 i$

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

$(6 + i) - (−2 - 4 i)$

$8 plus 5 i$

$(−2 + 5 i) - (−5 + 6 i)$

$(5 - \sqrt{−36}) + (2 - \sqrt{−49})$

$7 minus 13 i$

$(−3 + \sqrt{−64}) + (5 - \sqrt{−16})$

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

$25 - 2 \sqrt{2} i$

$(−5 + \sqrt{−27}) - (−4 - \sqrt{−48})$

Multiply Complex Numbers

In the following exercises, multiply.

$4 i open parenthesis 5 minus 3 i close parenthesis$

$12 plus 20 i$

$2 i open parenthesis −3 plus 4 i close parenthesis$

$−6 i open parenthesis −3 minus 2 i close parenthesis$

$−12 plus 18 i$

$− i open parenthesis 6 plus 5 i close parenthesis$

$(4 + 3 i) (−5 + 6 i)$

$−38 + + 9 i$

$(−2 - 5 i) (−4 + 3 i)$

$(−3 + 3 i) (−2 - 7 i)$

$27 plus 15 i$

$(−6 - 2 i) (−3 - 5 i)$

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

$open parenthesis 3 plus 4 i close parenthesis squared$

$−7 plus 24 i$

$open parenthesis −1 plus 5 i close parenthesis squared$

$open parenthesis −2 minus 3 i close parenthesis squared$

$−5 plus 12 i$

$open parenthesis −6 minus 5 i close parenthesis squared$

In the following exercises, multiply.

$\sqrt{−25} \cdot \sqrt{−36}$

$−30$

$\sqrt{−4} \cdot \sqrt{−16}$

$\sqrt{−9} \cdot \sqrt{−100}$

$−30$

$\sqrt{−64} \cdot \sqrt{−9}$

$(−2 - \sqrt{−27}) (4 - \sqrt{−48})$

$−44 - 4 i \sqrt{3}$

$(5 - \sqrt{−12}) (−3 + \sqrt{−75})$

$(2 + \sqrt{−8}) (−4 + \sqrt{−18})$

$−20 - 2 \sqrt{2} i$

$(5 + \sqrt{−18}) (−2 - \sqrt{−50})$

$(2 - i) (2 + i)$

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

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

$(−3 - 8 i) (−3 + 8 i)$

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.

$i to the power 41$

$i to the power 39$

$i to the power 66$

$−1$

$i to the power 48$

$i to the power 128$

$i to the power 162$

$i to the power 137$

$i to the power 255$

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.

The table has 4 columns and 4 rows. The first row is a header row with the headers “I can…”, “Confidently”, “With some help.”, and “No – I don’t get it!”. The first column contains the phrases “evaluate the square root of a negative number”, “add or subtract complex numbers”, “multiply complex numbers”, “divide complex numbers”, and “simplify powers of i”. The other columns are left blank so the learner can indicate their level of understanding.

ⓑ On a scale of $1 minus 10,$ 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.

ⓐ $the square root of 225$ ⓑ $− the square root of 16$

ⓐ 15 ⓑ $−4$

ⓐ $− the square root of 169$ ⓑ $the square root of −8$

ⓐ $the cube root of 8$ ⓑ $the 4th root of 81$ ⓒ $the 5th root of 243$

ⓐ 2 ⓑ 3 ⓒ 3

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

Estimate and Approximate Roots

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

ⓐ $the square root of 68$ ⓑ $the cube root of 84$

ⓐ $8 less than the square root of 68 less than 9$
ⓑ $4 less than the cube root of 84 less than 5$

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

ⓐ $the square root of 37$ ⓑ $the cube root of 84$ ⓒ $the 4th root of 125$

Simplify Variable Expressions with Roots

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

ⓐ $\sqrt[3]{a^{3}}$
ⓑ $\sqrt[7]{b^{7}}$

ⓐ a ⓑ $b$

ⓐ $\sqrt{a^{14}}$
ⓑ $\sqrt{w^{24}}$

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

ⓐ $m^{2}$ ⓑ $n to the power 4$

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

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

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

ⓐ $the square root of 144 x squared y squared$
ⓑ $\sqrt{169 w^{8} y^{10}}$
ⓒ $\sqrt[3]{8 a^{51} b^{6}}$

Simplify Radical Expressions

Use the Product Property to Simplify Radical Expressions

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

$the square root of 125$

$5 the square root of 5$

$the square root of 675$

ⓐ $the cube root of 625$ ⓑ $the 6th root of 128$

ⓐ $5 the cube root of 5$ ⓑ $2 the 6th root of 2$

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

ⓐ $\sqrt{a^{23}}$
ⓑ $\sqrt[3]{b^{8}}$
ⓒ $\sqrt[8]{c^{13}}$

ⓐ $\sqrt{80 s^{15}}$
ⓑ $\sqrt[5]{96 a^{7}}$
ⓒ $\sqrt[6]{128 b^{7}}$

ⓐ $4 | s^{7} | \sqrt[]{5 s}$ ⓑ $2 a the 5th root of 3 a squared$
ⓒ $2 | b | the 6th root of 2 b$

ⓐ $the square root of 96 r cubed s cubed$
ⓑ $\sqrt[3]{80 x^{7} y^{6}}$
ⓒ $\sqrt[4]{80 x^{8} y^{9}}$

ⓐ $the 5th root of −32$
ⓑ $the 8th root of −1$

ⓐ $−2$ ⓑ 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.

ⓐ $\sqrt{\frac{72}{98}}$ ⓑ $\sqrt[3]{\frac{24}{81}}$ ⓒ $\sqrt[4]{\frac{6}{96}}$

ⓐ $\frac{6}{7}$ ⓑ $\frac{2}{3}$ ⓒ $\frac{1}{2}$

ⓐ $\sqrt{\frac{y^{4}}{y^{8}}}$ ⓑ $\sqrt[5]{\frac{u^{21}}{u^{11}}}$ ⓒ $\sqrt[6]{\frac{v^{30}}{v^{12}}}$

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

$\frac{5 m^{2} \sqrt{3 m}}{4}$

ⓐ $\sqrt{\frac{28 p^{7}}{q^{2}}}$
ⓑ $\sqrt[3]{\frac{81 s^{8}}{t^{3}}}$
ⓒ $\sqrt[4]{\frac{64 p^{15}}{q^{12}}}$

ⓐ $\sqrt{\frac{27 p^{2} q}{108 p^{4} q^{3}}}$
ⓑ $\sqrt[3]{\frac{16 c^{5} d^{7}}{250 c^{2} d^{2}}}$
ⓒ $\sqrt[6]{\frac{2 m^{9} n^{7}}{128 m^{3} n}}$

ⓐ $\frac{1}{2 | p q |}$ ⓑ $\frac{2 c d}{5} \sqrt[3]{d^{2}}$
ⓒ $\frac{| m n |}{2}$

ⓐ $\frac{\sqrt{80 q^{5}}}{\sqrt{5 q}}$
ⓑ $\frac{\sqrt[3]{- 625}}{\sqrt[3]{5}}$
ⓒ $\frac{\sqrt[4]{80 m^{7}}}{\sqrt[4]{5 m}}$

Simplify Rational Exponents

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

In the following exercises, write as a radical expression.

ⓐ $r^{\frac{1}{2}}$ ⓑ $s^{\frac{1}{3}}$ ⓒ $t^{\frac{1}{4}}$

ⓐ $\sqrt{r}$ ⓑ $\sqrt[3]{s}$ ⓒ $the 4th root of t$

In the following exercises, write with a rational exponent.

ⓐ $the square root of 21 p$ ⓑ $the 4th root of 8 q$ ⓒ $4 the 6th root of 36 r$

In the following exercises, simplify.

ⓐ $625^{\frac{1}{4}}$
ⓑ $243^{\frac{1}{5}}$
ⓒ $32^{\frac{1}{5}}$

ⓐ 5 ⓑ 3 ⓒ 2

ⓐ ${(−1,000)}^{\frac{1}{3}}$
ⓑ $− {1,000}^{\frac{1}{3}}$
ⓒ ${(1,000)}^{- \frac{1}{3}}$

ⓐ ${(−32)}^{\frac{1}{5}}$
ⓑ ${(243)}^{- \frac{1}{5}}$
ⓒ $− 125^{\frac{1}{3}}$

ⓐ $−2$ ⓑ $\frac{1}{3}$ ⓒ $−5$

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

In the following exercises, write with a rational exponent.

ⓐ $\sqrt[4]{r^{7}}$
ⓑ $open parenthesis the 5th root of 2 p q close parenthesis cubed$
ⓒ $\sqrt[4]{{(\frac{12 m}{7 n})}^{3}}$

In the following exercises, simplify.

ⓐ $25^{\frac{3}{2}}$
ⓑ $9^{- \frac{3}{2}}$
ⓒ ${(−64)}^{\frac{2}{3}}$

ⓐ 125 ⓑ $\frac{1}{27}$ ⓒ 16

ⓐ $− 64^{\frac{3}{2}}$
ⓑ $− 64^{- \frac{3}{2}}$
ⓒ ${(−64)}^{\frac{3}{2}}$

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

ⓐ $6^{\frac{5}{2}} \cdot 6^{\frac{1}{2}}$
ⓑ ${(b^{15})}^{\frac{3}{5}}$
ⓒ $\frac{w^{\frac{2}{7}}}{w^{\frac{9}{7}}}$

ⓐ $6 cubed$ ⓑ $b to the power 9$ ⓒ $\frac{1}{w}$

ⓐ $\frac{a^{\frac{3}{4}} \cdot a^{- \frac{1}{4}}}{a^{- \frac{10}{4}}}$
ⓑ ${(\frac{27 b^{\frac{2}{3}} c^{- \frac{5}{2}}}{b^{- \frac{7}{3}} c^{\frac{1}{2}}})}^{\frac{1}{3}}$

Add, Subtract and Multiply Radical Expressions

Add and Subtract Radical Expressions

In the following exercises, simplify.

ⓐ $7 \sqrt{2} - 3 \sqrt{2}$
ⓑ $7 \sqrt[3]{p} + 2 \sqrt[3]{p}$
ⓒ $5 \sqrt[3]{x} - 3 \sqrt[3]{x}$

ⓐ $4 the square root of 2$ ⓑ $9 the cube root of p$ ⓒ $2 the cube root of x$

ⓐ $\sqrt{11 b} - 5 \sqrt{11 b} + 3 \sqrt{11 b}$
ⓑ $8 \sqrt[4]{11 c d} + 5 \sqrt[4]{11 c d} - 9 \sqrt[4]{11 c d}$

ⓐ $\sqrt{48} + \sqrt{27}$
ⓑ $\sqrt[3]{54} + \sqrt[3]{128}$
ⓒ $6 \sqrt[4]{5} - \frac{3}{2} \sqrt[4]{80}$

ⓐ $7 the square root of 3$ ⓑ $7 the cube root of 2$ ⓒ $3 the 4th root of 5$

ⓐ $\sqrt{80 c^{7}} - \sqrt{20 c^{7}}$
ⓑ $2 \sqrt[4]{162 r^{10}} + 4 \sqrt[4]{32 r^{10}}$

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

$37 y the square root of 3$

Multiply Radical Expressions

In the following exercises, simplify.

ⓐ $(5 \sqrt{6}) (− \sqrt{12})$
ⓑ $(−2 \sqrt[4]{18}) (− \sqrt[4]{9})$

ⓐ $(3 \sqrt{2 x^{3}}) (7 \sqrt{18 x^{2}})$
ⓑ $(−6 \sqrt[3]{20 a^{2}}) (−2 \sqrt[3]{16 a^{3}})$

ⓐ $126 x squared the square root of x$ ⓑ $48 a the cube root of 5 a squared$

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.

ⓐ $\frac{\sqrt{48}}{\sqrt{75}}$
ⓑ $\frac{\sqrt[3]{81}}{\sqrt[3]{24}}$

ⓐ $\frac{\sqrt{320 m n^{−5}}}{\sqrt{45 m^{−7} n^{3}}}$
ⓑ $\frac{\sqrt[3]{16 x^{4} y^{−2}}}{\sqrt[3]{−54 x^{−2} y^{4}}}$

ⓐ $\frac{8 m^{4}}{3 n^{4}}$ ⓑ $- \frac{2 x^{2}}{3 y^{2}}$

Rationalize a One Term Denominator

In the following exercises, rationalize the denominator.

ⓐ $\frac{8}{\sqrt{3}}$ ⓑ $\sqrt{\frac{7}{40}}$ ⓒ $\frac{8}{\sqrt{2 y}}$

ⓐ $\frac{1}{\sqrt[3]{11}}$ ⓑ $\sqrt[3]{\frac{7}{54}}$ ⓒ $\frac{3}{\sqrt[3]{3 x^{2}}}$

ⓐ $\frac{\sqrt[3]{121}}{11}$ ⓑ $\frac{\sqrt[3]{28}}{6}$ ⓒ $\frac{\sqrt[3]{9 x}}{x}$

ⓐ $\frac{1}{\sqrt[4]{4}}$ ⓑ $\sqrt[4]{\frac{9}{32}}$ ⓒ $\frac{6}{\sqrt[4]{9 x^{3}}}$

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

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

$\sqrt{u - 3} + 3 = u$

$u = 3, u = 4$

$\sqrt[3]{4 x + 5} - 2 = −5$

${(8 x + 5)}^{\frac{1}{3}} + 2 = −1$

$x equals −4$

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

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

$r equals 3$

Solve Radical Equations with Two Radicals

In the following exercises, solve.

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

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

$x = −8, x = 2$

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

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

$x equals 3$

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
ⓐ $g open parenthesis 4 close parenthesis$
ⓑ $g open parenthesis 8 close parenthesis$

$G (x) = \sqrt{5 x - 1},$ find
ⓐ $G open parenthesis 5 close parenthesis$
ⓑ $G open parenthesis 2 close parenthesis$

ⓐ $G (5) = 2 \sqrt{6}$ ⓑ $G open parenthesis 2 close parenthesis equals 3$

$h (x) = \sqrt[3]{x^{2} - 4},$ find
ⓐ $h open parenthesis −2 close parenthesis$
ⓑ $h open parenthesis 6 close parenthesis$

For the function
$g (x) = \sqrt[4]{4 - 4 x},$ find
ⓐ $g open parenthesis 1 close parenthesis$
ⓑ $g open parenthesis −3 close parenthesis$

ⓐ $g open parenthesis 1 close parenthesis equals 0$ ⓑ $g open parenthesis −3 close parenthesis equals 2$

Find the Domain of a Radical Function

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

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

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

$open parenthesis 2, ∞ close parenthesis$

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

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

$(- \infty, \frac{10}{7}]$

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: $[ 0, ∞ close parenthesis$
ⓑ

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).

[ 0, ∞ close parenthesis

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

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

ⓐ domain: $open parenthesis − ∞, ∞ close parenthesis$
ⓑ

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).

open parenthesis − ∞, ∞ close parenthesis

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.

Add or Subtract Complex Numbers

In the following exercises, add or subtract.

$\sqrt{−50} + \sqrt{−18}$

$8 the square root of 2 i$

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

$(6 + i) - (−2 - 4 i)$

$8 plus 5 i$

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

Multiply Complex Numbers

In the following exercises, multiply.

$(−2 - 5 i) (−4 + 3 i)$

$23 plus 14 i$

$−6 i open parenthesis −3 minus 2 i close parenthesis$

$\sqrt{−4} \cdot \sqrt{−16}$

$−8$

$(5 - \sqrt{−12}) (−3 + \sqrt{−75})$

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

$open parenthesis −2 minus 3 i close parenthesis squared$

$−5 plus 12 i$

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.

$i to the power 48$

$i to the power 255$

Practice Test

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

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

$5 x cubed$

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

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

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

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

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

ⓐ $256^{- \frac{1}{4}}$ ⓑ $− 49^{\frac{3}{2}}$

ⓐ $\frac{1}{4}$ ⓑ $−343$

$the square root of −45$

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

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

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

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

$− x squared the square root of 3 x$

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

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

$36 x^{4} \sqrt{2}$

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

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

$2 - 7 \sqrt{3}$

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

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

$\frac{7 x^{2} \sqrt{x}}{3 | y^{3} | \sqrt{y}}$

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

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

$3 (2 - \sqrt{3})$

$\sqrt{−4} \cdot \sqrt{−9}$

$−4 i open parenthesis −2 minus 3 i close parenthesis$

$−12 plus 8 i$

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

$i to the power 172$

$1$

In the following exercises, solve.

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

$\sqrt{x + 5} + 1 = x$

$x equals 4$

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

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: $[ −2, ∞ close parenthesis$
ⓑ

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).

[ 0, ∞ close parenthesis

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

	$2 i open parenthesis 7 minus 5 i close parenthesis$
Distribute.	$14 i minus 10 i squared$
Simplify $i^{2} .$	$14 i minus 10 open parenthesis −1 close parenthesis$
Multiply.	$14 i plus 10$
Write in standard form.	$10 plus 14 i$

	$(3 + 2 i) (4 - 3 i)$
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.	$18 minus i$

	$A mathematical expression showing the fraction (a + b) / (3 + 2i) squared. The numerator 'a + b' is displayed in red text.$
Use the Product of Binomial Squares Pattern, ${(a + b)}^{2} = a^{2} + 2 a b + b^{2} .$
Simplify.
Simplify $i^{2} .$
Simplify.

	$i to the power 86$
Divide 86 by 4 and rewrite $i to the power 86$ in the $i^{n} = {(i^{4})}^{q} \cdot i^{r}$ form.	${(1^{4})}^{21} \cdot i^{2}$

Simplify.	${(1)}^{21} \cdot (−1)$
Simplify.	$–1$

	$(3 - 2 i) (3 + 2 i)$
Use FOIL.	$9 + 6 i - 6 i - 4 i^{2}$
Combine like terms and simplify $i^{2} .$	$9 minus 4 open parenthesis −1 close parenthesis$
Multiply and combine like terms.	13


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