Skip to main content
permalink

Section 6.5 Radical Expressions and Equations Chapter Review

permalink

Subsection 6.5.1 Square and nth Root Properties

permalinkIn Section 6.1 we defined the square root x and nth root xn radicals. When x is positive, the expression xn means a positive number r, where rβ‹…rβ‹…β‹―β‹…r⏞n times=x. The square root x is just the case where n=2.

permalinkWhen x is negative, xn might not be defined. It depends on whether or not n is an even number. When x is negative and n is odd, xn is a negative number where rβ‹…rβ‹…β‹―β‹…r⏞n times=x.

permalinkThere are two helpful rules for simplifying radicals.

permalink
List 6.5.1. Rules of Radicals for Multiplication and Division

If a and b are positive real numbers, and m is a positive integer , then we have the following rules:

Root of a Product Rule

aβ‹…bm=amβ‹…bm

Root of a Quotient Rule

abm=ambm as long as b≠0

permalink
Checkpoint 6.5.2.
permalink

Subsection 6.5.2 Rationalizing the Denominator

permalinkIn Section 6.2 we covered how to rationalize the denominator when it contains a single square root or a binomial with a square root term.

permalink
Example 6.5.3.

Rationalize the denominator of the expressions.

  1. 123

  2. 575

Explanation
  1. \begin{align*} \frac{12}{\sqrt{3}}\amp=\frac{12}{\sqrt{3}}\multiplyright{\frac{\sqrt{3}}{\sqrt{3}}}\\ \amp=\frac{12\sqrt{3}}{3}\\ \amp=4\sqrt{3} \end{align*}
  2. First we will simplify \(\sqrt{75}\text{.}\)

    \begin{align*} \frac{\sqrt{5}}{\sqrt{75}}\amp=\frac{\sqrt{5}}{\sqrt{25\cdot 3}}\\ \amp=\frac{\sqrt{5}}{\sqrt{25}\cdot\sqrt{3}}\\ \amp=\frac{\sqrt{5}}{5\sqrt{3}}\\ \end{align*}

    Now we can rationalize the denominator by multiplying the numerator and denominator by \(\sqrt{3}\text{.}\)

    \begin{align*} \amp=\frac{\sqrt{5}}{5\sqrt{3}}\multiplyright{\frac{\sqrt{3}}{\sqrt{3}}}\\ \amp=\frac{\sqrt{15}}{5\cdot 3}\\ \amp=\frac{\sqrt{15}}{15} \end{align*}
permalink
Example 6.5.4. Rationalize Denominator Using the Difference of Squares Formula.

Rationalize the denominator in 6βˆ’53+2.

Explanation

To remove radicals in \(\sqrt{3}+\sqrt{2}\) with the difference of squares formula, we multiply it with \(\sqrt{3}-\sqrt{2}\text{.}\)

\begin{align*} \frac{\sqrt{6}-\sqrt{5}}{\sqrt{3}+\sqrt{2}}\amp=\frac{\sqrt{6}-\sqrt{5}}{\sqrt{3}+\sqrt{2}}\multiplyright{\frac{\left(\sqrt{3}-\sqrt{2}\right)}{\left(\sqrt{3}-\sqrt{2}\right)}}\\ \amp=\frac{\sqrt{6}\multiplyright{\sqrt{3}}-\sqrt{6}\multiplyright{\sqrt{2}}-\sqrt{5}\multiplyright{\sqrt{3}}-\sqrt{5}\multiplyright{-\sqrt{2}}}{\left(\sqrt{3}\right)^2-\left(\sqrt{2}\right)^2}\\ \amp=\frac{\sqrt{18}-\sqrt{12}-\sqrt{15}+\sqrt{10}}{3-2}\\ \amp=\frac{3\sqrt{2}-2\sqrt{3}-\sqrt{15}+\sqrt{10}}{1}\\ \amp=3\sqrt{2}-2\sqrt{3}-\sqrt{15}+\sqrt{10} \end{align*}
permalink

Subsection 6.5.3 Radical Expressions and Rational Exponents

permalinkIn Section 6.3 we learned the rational exponent rule and added it to our list of exponent rules.

permalink
Example 6.5.5. Radical Expressions and Rational Exponents.

Simplify the expressions using Fact 6.3.2 or Fact 6.3.9.

  1. 10012

  2. (βˆ’64)βˆ’13

  3. βˆ’8134

  4. (βˆ’127)23

Explanation
  1. \(\begin{aligned}[t] 100^{\sfrac{1}{2}}\amp=\left(\sqrt{100}\right)\\ \amp=10 \end{aligned}\)

  2. \(\begin{aligned}[t] (-64)^{-\sfrac{1}{3}}\amp=\frac{1}{(-64)^{\sfrac{1}{3}}}\\ \amp=\frac{1}{\left(\sqrt[3]{(-64)}\right)}\\ \amp=\frac{1}{-4} \end{aligned}\)

  3. \(\begin{aligned}[t] -81^{\sfrac{3}{4}}\amp=-\left(\sqrt[4]{81}\right)^3\\ \amp=-3^3\\ \amp=-27 \end{aligned}\)

  4. In this problem the negative can be associated with either the numerator or the denominator, but not both. We choose the numerator.

    \begin{align*} \left(-\frac{1}{27}\right)^{\sfrac{2}{3}}\amp=\left(\sqrt[3]{-\frac{1}{27}}\right)^2\\ \amp=\left(\frac{\sqrt[3]{-1}}{\sqrt[3]{27}}\right)^2\\ \amp=\left(\frac{-1}{3}\right)^2\\ \amp=\frac{(-1)^2}{(3)^2}\\ \amp=\frac{1}{9} \end{align*}
permalink
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.

  1. 7z59

  2. 54xβˆ’23

  3. (βˆ’9q5)45

  4. y5β‹…y24

  5. t3t23

  6. x3

  7. 5(4+a12)2

  8. βˆ’6(2pβˆ’52)35

Explanation
  1. \(\begin{aligned}[t] 7z^{\sfrac{5}{9}}\amp=7\sqrt[9]{z^5} \end{aligned}\)

  2. \(\begin{aligned}[t] \frac{5}{4}x^{-\sfrac{2}{3}}\amp=\frac{5}{4}\cdot\frac{1}{x^{\sfrac{2}{3}}}\\ \amp=\frac{5}{4}\cdot\frac{1}{\sqrt[3]{x^2}}\\ \amp=\frac{5}{4\sqrt[3]{x^2}} \end{aligned}\)

  3. \(\begin{aligned}[t] \left(-9q^5\right)^{\sfrac{4}{5}}\amp=\left(-9\right)^{\sfrac{4}{5}}\cdot\left(q^5\right)^{\sfrac{4}{5}}\\ \amp=\left(-9\right)^{\sfrac{4}{5}}\cdot q^{5\cdot\sfrac{4}{5}}\\ \amp=\left(\sqrt[5]{-9}\right)^4\cdot q^{4}\\ \amp=\left(q\sqrt[5]{-9}\right)^4 \end{aligned}\)

  4. \(\begin{aligned}[t] \sqrt{y^5}\cdot\sqrt[4]{y^2}\amp=y^{\sfrac{5}{2}}\cdot y^{\sfrac{2}{4}}\\ \amp=y^{\sfrac{5}{2}+\sfrac{2}{4}}\\ \amp=y^{\sfrac{10}{4}+\sfrac{2}{4}}\\ \amp=y^{\sfrac{12}{4}}\\ \amp=y^3 \end{aligned}\)

  5. \(\begin{aligned}[t] \frac{\sqrt{t^3}}{\sqrt[3]{t^2}}\amp=\frac{t^{\sfrac{3}{2}}}{t^{\sfrac{2}{3}}}\\ \amp=t^{\sfrac{3}{2}-\sfrac{2}{3}}\\ \amp=t^{\sfrac{9}{6}-\sfrac{4}{6}}\\ \amp=t^{\sfrac{5}{6}}\\ \amp=\sqrt[6]{t^5} \end{aligned}\)

  6. \(\begin{aligned}[t] \sqrt{\sqrt[3]{x}}\amp=\sqrt{x^{\sfrac{1}{3}}}\\ \amp=\left(x^{\sfrac{1}{3}}\right)^{\sfrac{1}{2}}\\ \amp=x^{\sfrac{1}{3}\cdot\sfrac{1}{2}}\\ \amp=x^{\sfrac{1}{6}}\\ \amp=\sqrt[6]{x} \end{aligned}\)

  7. \(\begin{aligned}[t] 5\left(4+a^{\sfrac{1}{2}}\right)^2\amp=5\left(4+a^{\sfrac{1}{2}}\right)\left(4+a^{\sfrac{1}{2}}\right)\\ \amp=5\left(4^2+2\cdot4\cdot a^{\sfrac{1}{2}}+\left(a^{\sfrac{1}{2}}\right)^2\right)\\ \amp=5\left(16+8a^{\sfrac{1}{2}}+a^{\sfrac{1}{2}\cdot 2}\right)\\ \amp=5\left(16+8a^{\sfrac{1}{2}}+a\right)\\ \amp=5\left(16+8\sqrt{a}+a\right)\\ \amp=80+40\sqrt{a}+5a \end{aligned}\)

  8. \(\begin{aligned}[t] -6\left(2p^{-\sfrac{5}{2}}\right)^{\sfrac{3}{5}}\amp=-6\cdot2^{\sfrac{3}{5}}\cdot p^{-\sfrac{5}{2}\cdot\sfrac{3}{5}}\\ \amp=-6\cdot2^{\sfrac{3}{5}}\cdot p^{-\sfrac{3}{2}}\\ \amp=-\frac{6\cdot 2^{\sfrac{3}{5}}}{p^{\sfrac{3}{2}}}\\ \amp=-\frac{6\sqrt[5]{2^3}}{\sqrt{p^3}}\\ \amp=-\frac{6\sqrt[5]{8}}{\sqrt{p^3}} \end{aligned}\)

permalink

Subsection 6.5.4 Solving Radical Equations

permalinkIn Section 6.4 we covered solving equations that contain a radical. We learned about extraneous solutions and the need to check our solutions.

permalink
Example 6.5.7. Solving Radical Equations.

Solve for r in r=9+r+3.

Explanation

We will isolate the radical first, and then square both sides.

\begin{align*} r\amp=9+\sqrt{r+3}\\ r-9\amp=\sqrt{r+3}\\ \left(r-9\right)^{\highlight{2}}\amp=\left(\sqrt{r+3}\right)^{\highlight{2}}\\ r^2-18r+81\amp=r+3\\ r^2-19r+78\amp=0\\ (r-6)(r-13)\amp=0 \end{align*}
\begin{align*} r-6\amp=0\amp\amp\text{ or }r-13\amp=0\\ r\amp=6\amp\amp\text{ or }r\amp=13 \end{align*}

Because we squared both sides of an equation, we must check both solutions.

\begin{align*} \substitute{6}\amp\stackrel{?}{=}9+\sqrt{\substitute{6}+3}\amp\substitute{13}\amp\stackrel{?}{=}9+\sqrt{\substitute{13}+3}\\ 6\amp\stackrel{?}{=}9+\sqrt{9}\amp13\amp\stackrel{?}{=}9+\sqrt{16}\\ 6\amp\stackrel{\text{no}}{=}9+3\amp13\amp\stackrel{\checkmark}{=}9+4 \end{align*}

It turns out \(6\) is an extraneous solution and \(13\) is a valid solution. So the equation has one solution: \(13\text{.}\) The solution set is \(\{13\}\text{.}\)

permalink
Example 6.5.8. Solving Radical Equations that Require Squaring Twice.

Solve the equation t+9=βˆ’1βˆ’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{align*} \sqrt{t+9}\amp=-1-\sqrt{t}\\ \left(\sqrt{t+9}\right)^{\highlight{2}}\amp=\left(-1-\sqrt{t}\right)^{\highlight{2}}\\ t+9\amp=1+2\sqrt{t}+t \amp\text{ after expanding the binomial squared}\\ 9\amp=1+2\sqrt{t}\\ 8\amp=2\sqrt{t}\\ 4\amp=\sqrt{t}\\ (4)^{\highlight{2}}\amp=\left(\sqrt{t}\right)^{\highlight{2}}\\ 16\amp=t \end{align*}

Because we squared both sides of an equation, we must check the solution by substituting \(\substitute{16}\) into \(\sqrt{t+9}=-1-\sqrt{t}\text{,}\) and we have:

\begin{align*} \sqrt{t+9}\amp=-1-\sqrt{t}\\ \sqrt{\substitute{16}+9}\amp\stackrel{?}{=}-1-\sqrt{16}\\ \sqrt{25}\amp\stackrel{?}{=}-1-4\\ 5\amp\stackrel{\text{no}}{=}-5 \end{align*}

Our solution did not check so there is no solution to this equation. The solution set is the empty set, which can be denoted \(\{\text{ }\}\) or \(\emptyset\text{.}\)

permalink

Exercises 6.5.5 Exercises

Square Root and nth Root
permalink
1.

Evaluate the following.

1100=.

permalink
2.

Evaluate the following.

4121=.

permalink
3.

Evaluate the following.

βˆ’16=.

permalink
4.

Evaluate the following.

βˆ’25=.

permalink
5.

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

483=

permalink
6.

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

322=

permalink
7.

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

250=

permalink
8.

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

99=

permalink
9.

Simplify the expression.

913β‹…9121=

permalink
10.

Simplify the expression.

93β‹…74=

permalink
11.

Simplify the expression.

52β‹…72=

permalink
12.

Simplify the expression.

73β‹…13=

permalink
13.

Simplify the expression.

1310βˆ’1410=

permalink
14.

Simplify the expression.

145βˆ’155=

permalink
15.

Simplify the expression.

180+45=

permalink
16.

Simplify the expression.

80+125=

permalink
17.

Simplify 646.

permalink
18.

Simplify 643.

permalink
19.

Simplify βˆ’83.

permalink
20.

Simplify βˆ’83.

permalink
21.

Simplify βˆ’164.

permalink
22.

Simplify βˆ’814.

permalink
23.

Simplify 1444.

permalink
24.

Simplify 1353.

permalink
25.

Simplify 1183.

permalink
26.

Simplify 9646.

permalink
27.

Simplify 40273.

permalink
28.

Simplify 561253.

Rationalizing the Denominator
permalink
29.

Rationalize the denominator and simplify the expression.

2252=

permalink
30.

Rationalize the denominator and simplify the expression.

6112=

permalink
31.

Rationalize the denominator and simplify the expression.

227=

permalink
32.

Rationalize the denominator and simplify the expression.

5112=

permalink
33.

Rationalize the denominator and simplify the expression.

615+8=

permalink
34.

Rationalize the denominator and simplify the expression.

77+4=

permalink
35.

Rationalize the denominator and simplify the expression.

5βˆ’1313+3=

permalink
36.

Rationalize the denominator and simplify the expression.

3βˆ’147+10=

Radical Expressions and Rational Exponents
permalink
37.

Without using a calculator, evaluate the expression.

125βˆ’23=

permalink
38.

Without using a calculator, evaluate the expression.

8βˆ’53=

permalink
39.

Without using a calculator, evaluate the expression.

(181)βˆ’34=

permalink
40.

Without using a calculator, evaluate the expression.

(19)βˆ’32=

permalink
41.

Without using a calculator, evaluate the expression.

12523=

permalink
42.

Without using a calculator, evaluate the expression.

8134=

permalink
43.

Without using a calculator, evaluate the expression.

10245=

permalink
44.

Without using a calculator, evaluate the expression.

643=

permalink
45.

Use rational exponents to write the expression.

b5=

permalink
46.

Use rational exponents to write the expression.

c=

permalink
47.

Use rational exponents to write the expression.

8x+75=

permalink
48.

Use rational exponents to write the expression.

5z+14=

permalink
49.

Convert the expression to radical notation.

t23 =

permalink
50.

Convert the expression to radical notation.

r45 =

permalink
51.

Convert the expression to radical notation.

m54 =

permalink
52.

Convert the expression to radical notation.

r23 =

permalink
53.

Convert the expression to radical notation.

515a45 =

permalink
54.

Convert the expression to radical notation.

1314b34 =

permalink
55.

Simplify the expression, answering with rational exponents and not radicals.

c11c11=

permalink
56.

Simplify the expression, answering with rational exponents and not radicals.

x9x9=

permalink
57.

Simplify the expression, answering with rational exponents and not radicals.

32z25=

permalink
58.

Simplify the expression, answering with rational exponents and not radicals.

125t53=

permalink
59.

Simplify the expression, answering with rational exponents and not radicals.

16rr310=

permalink
60.

Simplify the expression, answering with rational exponents and not radicals.

36mm310=

permalink
61.

Simplify the expression, answering with rational exponents and not radicals.

nβ‹…n56=

permalink
62.

Simplify the expression, answering with rational exponents and not radicals.

aβ‹…a310=

Solving Radical Equations
permalink
63.

Solve the equation.

t=tβˆ’3+5

permalink
64.

Solve the equation.

t=tβˆ’1+3

permalink
65.

Solve the equation.

x+9=x+1

permalink
66.

Solve the equation.

x+8=x+2

permalink
67.

Solve the equation.

y+110=y

permalink
68.

Solve the equation.

y+56=y

permalink
69.

Solve the equation.

r=r+4+16

permalink
70.

Solve the equation.

r=r+2+88

permalink
71.

Solve the equation.

52βˆ’t=t+4

permalink
72.

Solve the equation.

17βˆ’t=t+3

permalink
73.

According to the Pythagorean Theorem, the length c of the hypothenuse of a rectangular triangle can be found through the following equation.

c=a2+b2

If a rectangular triangle has a hypothenuse of 41 ft and one leg is 40 ft long, how long is the third side of the triangle?

The third side of the triangle is long.

permalink
74.

According to the Pythagorean Theorem, the length c of the hypothenuse of a rectangular triangle can be found through the following equation.

c=a2+b2

If a rectangular triangle has a hypothenuse of 17 ft and one leg is 15 ft long, how long is the third side of the triangle?

The third side of the triangle is long.

permalink
75.

A pendulum has the length L ft. The time period T that it takes to once swing back and forth is 2 s. Use the following formula to find its length.

T=2Ο€L32

The pendulum is long.

permalink
76.

A pendulum has the length L ft. The time period T that it takes to once swing back and forth is 4 s. Use the following formula to find its length.

T=2Ο€L32

The pendulum is long.