ORCCA Radical Expressions and Equations Chapter Review

Section 6.5 Radical Expressions and Equations Chapter Review

Subsection 6.5.1 Square and $n$ th Root Properties

In Section 1 we defined the square root

\sqrt{x}

and

n

th root

\sqrt[n]{x}

radicals. When

x

is positive, the expression

\sqrt[n]{x}

means a positive number

r,

where

\overset{n times}{\overset{⏞}{r \cdot r \cdot \dots \cdot r}} = x .

The square root

\sqrt{x}

is just the case where

n = 2 .

When

x

is negative,

\sqrt[n]{x}

might not be defined. It depends on whether or not

n

is an even number. When

x

is negative and

n

is odd,

\sqrt[n]{x}

is a negative number where

\overset{n times}{\overset{⏞}{r \cdot r \cdot \dots \cdot r}} = x .

There are two helpful properties for simplifying radicals.

List 6.5.1. Properties of Radicals for Multiplication and Division

a

and

b

are positive real numbers, and

m

is a positive [cross-reference to target(s) "item-integer-definition" missing or not unique], then we have the following properties:

Root of a Product Property: $\sqrt[m]{a \cdot b} = \sqrt[m]{a} \cdot \sqrt[m]{b}$
Root of a Quotient Property: $\sqrt[m]{\frac{a}{b}} = \frac{\sqrt[m]{a}}{\sqrt[m]{b}}$ as long as $b \neq 0$

Checkpoint 6.5.2.

Simplify $\sqrt{72} .$
Simplify $\sqrt[3]{72} .$
Simplify $\sqrt{\frac{72}{25}} .$

Explanation.

$\begin{aligned} \sqrt{72} & = \sqrt{4 \cdot 18} \\ = \sqrt{4} \cdot \sqrt{18} \\ = 2 \sqrt{18} \\ = 2 \sqrt{9 \cdot 2} \\ = 2 \sqrt{9} \cdot \sqrt{2} \\ = 2 \cdot 3 \sqrt{2} \\ = 6 \sqrt{2} \end{aligned}$
$\begin{aligned} \sqrt[3]{72} & = \sqrt[3]{8 \cdot 9} \\ = \sqrt[3]{8} \cdot \sqrt[3]{9} \\ = 2 \sqrt[3]{9} \end{aligned}$
$\begin{aligned} \sqrt{\frac{72}{25}} & = \frac{\sqrt{72}}{\sqrt{25}} \\ = \frac{6 \sqrt{2}}{5} \end{aligned}$

Subsection 6.5.2 Rationalizing the Denominator

In Section 2 we covered how to rationalize the denominator when it contains a single square root or a binomial with a square root term.

Example 6.5.3.

Rationalize the denominator of the expressions.

$\frac{12}{\sqrt{3}}$
$\frac{\sqrt{5}}{\sqrt{75}}$

Explanation.

$\begin{aligned} \frac{12}{\sqrt{3}} & = \frac{12}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} \\ = \frac{12 \sqrt{3}}{3} \\ = 4 \sqrt{3} \end{aligned}$
First we will simplify $\sqrt{75} .$

$\begin{aligned} \frac{\sqrt{5}}{\sqrt{75}} & = \frac{\sqrt{5}}{\sqrt{25 \cdot 3}} \\ = \frac{\sqrt{5}}{\sqrt{25} \cdot \sqrt{3}} \\ = \frac{\sqrt{5}}{5 \sqrt{3}} \end{aligned}$
Now we can rationalize the denominator by multiplying the numerator and denominator by $\sqrt{3} .$
$\begin{aligned} = \frac{\sqrt{5}}{5 \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} \\ = \frac{\sqrt{15}}{5 \cdot 3} \\ = \frac{\sqrt{15}}{15} \end{aligned}$

Example 6.5.4. Rationalize Denominator Using the Difference of Squares Formula.

Rationalize the denominator in

\frac{\sqrt{6} - \sqrt{5}}{\sqrt{3} + \sqrt{2}} .

Explanation.

To remove radicals in

\sqrt{3} + \sqrt{2}

with the difference of squares formula, we multiply it with

\sqrt{3} - \sqrt{2} .

\begin{aligned} \frac{\sqrt{6} - \sqrt{5}}{\sqrt{3} + \sqrt{2}} & = \frac{\sqrt{6} - \sqrt{5}}{\sqrt{3} + \sqrt{2}} \cdot \frac{(\sqrt{3} - \sqrt{2})}{(\sqrt{3} - \sqrt{2})} \\ = \frac{\sqrt{6} \cdot \sqrt{3} - \sqrt{6} \cdot \sqrt{2} - \sqrt{5} \cdot \sqrt{3} - \sqrt{5} \cdot - \sqrt{2}}{{(\sqrt{3})}^{2} - {(\sqrt{2})}^{2}} \\ = \frac{\sqrt{18} - \sqrt{12} - \sqrt{15} + \sqrt{10}}{3 - 2} \\ = \frac{3 \sqrt{2} - 2 \sqrt{3} - \sqrt{15} + \sqrt{10}}{1} \\ = 3 \sqrt{2} - 2 \sqrt{3} - \sqrt{15} + \sqrt{10} \end{aligned}

Subsection 6.5.3 Radical Expressions and Rational Exponents

In Section 3 we learned the rational exponent property and added it to our list of exponent properties.

Example 6.5.5. Radical Expressions and Rational Exponents.

Simplify the expressions using Fact 6.3.2 or Fact 6.3.9.

$100^{1 / 2}$
$(- 64)^{- 1 / 3}$
$- 81^{3 / 4}$
${(- \frac{1}{27})}^{2 / 3}$

Explanation.

$\begin{aligned} 100^{1 / 2} & = (\sqrt{100}) \\ = 10 \end{aligned}$
$\begin{aligned} (- 64)^{- 1 / 3} & = \frac{1}{(- 64)^{1 / 3}} \\ = \frac{1}{(\sqrt[3]{(- 64)})} \\ = \frac{1}{- 4} \end{aligned}$
$\begin{aligned} - 81^{3 / 4} & = - {(\sqrt[4]{81})}^{3} \\ = - 3^{3} \\ = - 27 \end{aligned}$
In this problem the negative can be associated with either the numerator or the denominator, but not both. We choose the numerator.

$\begin{aligned} {(- \frac{1}{27})}^{2 / 3} & = {(\sqrt[3]{- \frac{1}{27}})}^{2} \\ = {(\frac{\sqrt[3]{- 1}}{\sqrt[3]{27}})}^{2} \\ = {(\frac{- 1}{3})}^{2} \\ = \frac{(- 1)^{2}}{(3)^{2}} \\ = \frac{1}{9} \end{aligned}$

Example 6.5.6. More Expressions with Rational Exponents.

Use exponent properties in List 6.3.14 to simplify the expressions, and write all final versions using radicals.

$7 z^{5 / 9}$
$\frac{5}{4} x^{- 2 / 3}$
${(- 9 q^{5})}^{4 / 5}$
$\sqrt{y^{5}} \cdot \sqrt[4]{y^{2}}$
$\frac{\sqrt{t^{3}}}{\sqrt[3]{t^{2}}}$
$\sqrt{\sqrt[3]{x}}$
$5 {(4 + a^{1 / 2})}^{2}$
$- 6 {(2 p^{- 5 / 2})}^{3 / 5}$

Explanation.

$\begin{aligned} 7 z^{5 / 9} & = 7 \sqrt[9]{z^{5}} \end{aligned}$
$\begin{aligned} \frac{5}{4} x^{- 2 / 3} & = \frac{5}{4} \cdot \frac{1}{x^{2 / 3}} \\ = \frac{5}{4} \cdot \frac{1}{\sqrt[3]{x^{2}}} \\ = \frac{5}{4 \sqrt[3]{x^{2}}} \end{aligned}$
$\begin{aligned} {(- 9 q^{5})}^{4 / 5} & = {(- 9)}^{4 / 5} \cdot {(q^{5})}^{4 / 5} \\ = {(- 9)}^{4 / 5} \cdot q^{5 \cdot 4 / 5} \\ = {(\sqrt[5]{- 9})}^{4} \cdot q^{4} \\ = {(q \sqrt[5]{- 9})}^{4} \end{aligned}$
$\begin{aligned} \sqrt{y^{5}} \cdot \sqrt[4]{y^{2}} & = y^{5 / 2} \cdot y^{2 / 4} \\ = y^{5 / 2 + 2 / 4} \\ = y^{10 / 4 + 2 / 4} \\ = y^{12 / 4} \\ = y^{3} \end{aligned}$
$\begin{aligned} \frac{\sqrt{t^{3}}}{\sqrt[3]{t^{2}}} & = \frac{t^{3 / 2}}{t^{2 / 3}} \\ = t^{3 / 2 - 2 / 3} \\ = t^{9 / 6 - 4 / 6} \\ = t^{5 / 6} \\ = \sqrt[6]{t^{5}} \end{aligned}$
$\begin{aligned} \sqrt{\sqrt[3]{x}} & = \sqrt{x^{1 / 3}} \\ = {(x^{1 / 3})}^{1 / 2} \\ = x^{1 / 3 \cdot 1 / 2} \\ = x^{1 / 6} \\ = \sqrt[6]{x} \end{aligned}$
$\begin{aligned} 5 {(4 + a^{1 / 2})}^{2} & = 5 (4 + a^{1 / 2}) (4 + a^{1 / 2}) \\ = 5 (4^{2} + 2 \cdot 4 \cdot a^{1 / 2} + {(a^{1 / 2})}^{2}) \\ = 5 (16 + 8 a^{1 / 2} + a^{1 / 2 \cdot 2}) \\ = 5 (16 + 8 a^{1 / 2} + a) \\ = 5 (16 + 8 \sqrt{a} + a) \\ = 80 + 40 \sqrt{a} + 5 a \end{aligned}$
$\begin{aligned} - 6 {(2 p^{- 5 / 2})}^{3 / 5} & = - 6 \cdot 2^{3 / 5} \cdot p^{- 5 / 2 \cdot 3 / 5} \\ = - 6 \cdot 2^{3 / 5} \cdot p^{- 3 / 2} \\ = - \frac{6 \cdot 2^{3 / 5}}{p^{3 / 2}} \\ = - \frac{6 \sqrt[5]{2^{3}}}{\sqrt{p^{3}}} \\ = - \frac{6 \sqrt[5]{8}}{\sqrt{p^{3}}} \end{aligned}$

Subsection 6.5.4 Solving Radical Equations

In Section 4 we covered solving equations that contain a radical. We learned about extraneous solutions and the need to check our solutions.

Example 6.5.7. Solving Radical Equations that Require Squaring Twice.

Solve the equation

\sqrt{t + 9} = - 1 - \sqrt{t}

for

t .

Explanation.

We cannot isolate two radicals, so we will simply square both sides, and later try to isolate the remaining radical.

\begin{aligned} \sqrt{t + 9} & = - 1 - \sqrt{t} \\ {(\sqrt{t + 9})}^{2} & = {(- 1 - \sqrt{t})}^{2} \\ t + 9 & = 1 + 2 \sqrt{t} + t & after expanding the binomial squared \\ 9 & = 1 + 2 \sqrt{t} \\ 8 & = 2 \sqrt{t} \\ 4 & = \sqrt{t} \\ (4)^{2} & = {(\sqrt{t})}^{2} \\ 16 & = t \end{aligned}

Because we squared both sides of an equation, we must check the solution by substituting

16

into

\sqrt{t + 9} = - 1 - \sqrt{t},

and we have:

\begin{aligned} \sqrt{t + 9} & = - 1 - \sqrt{t} \\ \sqrt{16 + 9} & \overset{?}{=} - 1 - \sqrt{16} \\ \sqrt{25} & \overset{?}{=} - 1 - 4 \\ 5 & \overset{no}{=} - 5 \end{aligned}

Our solution did not check so there is no solution to this equation. The solution set is the empty set, which can be denoted

{}

\emptyset .

Exercises 6.5.5 Exercises

Square Root and $n$ th Root.

1.

Evaluate the following.

\sqrt{\frac{16}{49}} =

2.

Evaluate the following.

\sqrt{\frac{25}{121}} =

3.

Evaluate the following.

- \sqrt{64} =

4.

Evaluate the following.

- \sqrt{81} =

Exercise Group.

5.

Simplify the radical expression or state that it is not a real number.

\frac{\sqrt{125}}{\sqrt{5}} =

6.

Simplify the radical expression or state that it is not a real number.

\frac{\sqrt{144}}{\sqrt{4}} =

7.

Simplify the radical expression or state that it is not a real number.

\sqrt{8} =

8.

Simplify the radical expression or state that it is not a real number.

\sqrt{98} =

Exercise Group.

9.

Simplify the expression.

3 \sqrt{7} \cdot 5 \sqrt{25} =

10.

Simplify the expression.

4 \sqrt{13} \cdot 2 \sqrt{121} =

11.

Simplify the expression.

\sqrt{\frac{5}{6}} \cdot \sqrt{\frac{1}{6}} =

12.

Simplify the expression.

\sqrt{\frac{2}{7}} \cdot \sqrt{\frac{5}{7}} =

Exercise Group.

13.

Simplify the expression.

17 \sqrt{2} - 18 \sqrt{2} =

14.

Simplify the expression.

18 \sqrt{11} - 19 \sqrt{11} =

15.

Simplify the expression.

\sqrt{63} + \sqrt{252} =

16.

Simplify the expression.

\sqrt{252} + \sqrt{63} =

Exercise Group.

17.

\sqrt[3]{64} .

18.

\sqrt[3]{27} .

19.

\sqrt[5]{- 32} .

20.

\sqrt[3]{- 125} .

21.

\sqrt[4]{- 16} .

22.

\sqrt[4]{- 81} .

23.

\sqrt[6]{128} .

24.

\sqrt[6]{128} .

25.

\sqrt[5]{\frac{7}{32}} .

26.

\sqrt[3]{\frac{7}{27}} .

27.

\sqrt[3]{\frac{135}{64}} .

28.

\sqrt[3]{\frac{135}{64}} .

Rationalizing the Denominator.

29.

Rationalize the denominator and simplify the expression.

\frac{4}{\sqrt{216}} =

30.

Rationalize the denominator and simplify the expression.

\frac{2}{\sqrt{80}} =

31.

Rationalize the denominator and simplify the expression.

\sqrt{\frac{7}{150}} =

32.

Rationalize the denominator and simplify the expression.

\sqrt{\frac{2}{125}} =

33.

Rationalize the denominator and simplify the expression.

\frac{9}{\sqrt{2} + 5} =

34.

Rationalize the denominator and simplify the expression.

\frac{9}{\sqrt{17} + 8} =

35.

Rationalize the denominator and simplify the expression.

\frac{\sqrt{3} - 6}{\sqrt{7} + 8} =

36.

Rationalize the denominator and simplify the expression.

\frac{\sqrt{2} - 7}{\sqrt{13} + 5} =

Radical Expressions and Rational Exponents.

37.

Without using a calculator, evaluate the expression.

128^{- \frac{4}{7}} =

38.

Without using a calculator, evaluate the expression.

9^{- \frac{1}{2}} =

39.

Without using a calculator, evaluate the expression.

{(\frac{1}{81})}^{- \frac{3}{4}} =

40.

Without using a calculator, evaluate the expression.

{(\frac{1}{9})}^{- \frac{3}{2}} =

41.

Without using a calculator, evaluate the expression.

\sqrt[5]{32^{2}} =

42.

Without using a calculator, evaluate the expression.

\sqrt[5]{32^{4}} =

43.

Without using a calculator, evaluate the expression.

\sqrt[5]{1024} =

44.

Without using a calculator, evaluate the expression.

\sqrt[3]{64} =

Exercise Group.

45.

Use rational exponents to write the expression.

\sqrt[9]{z}

46.

Use rational exponents to write the expression.

\sqrt[6]{t}

47.

Use rational exponents to write the expression.

\sqrt[3]{3 r + 3} =

48.

Use rational exponents to write the expression.

\sqrt{9 m + 8} =

Exercise Group.

49.

Convert the expression to radical notation.

n^{\frac{4}{5}}

50.

Convert the expression to radical notation.

a^{\frac{2}{3}}

51.

Convert the expression to radical notation.

b^{\frac{8}{9}}

52.

Convert the expression to radical notation.

r^{\frac{2}{3}}

53.

Convert the expression to radical notation.

16^{\frac{1}{3}} x^{\frac{2}{3}}

54.

Convert the expression to radical notation.

5^{\frac{1}{6}} z^{\frac{5}{6}}