Skip to main content
Logo image

Section 6.5 Radical Expressions and Equations Chapter Review

Subsection 6.5.1 Square and nth Root Properties

In Section 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 rrrn times=x. The square root x is just the case where n=2.
When 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 rrrn times=x.
There are two helpful properties for simplifying radicals.
List 6.5.1. Properties of Radicals for Multiplication and Division
If 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
abm=ambm
Root of a Quotient Property
abm=ambm as long as b0

Checkpoint 6.5.2.

  1. Simplify 72.
  2. Simplify 723.
  3. Simplify 7225.
Explanation.
  1.  
    72=418=418=218=292=292=232=62
  2.  
    723=893=8393=293
  3.  
    7225=7225=625

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.
  1. 123
  2. 575
Explanation.
  1. 123=12333=1233=43
  2. First we will simplify 75.
    575=5253=5253=553
    Now we can rationalize the denominator by multiplying the numerator and denominator by 3.
    =55333=1553=1515

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

Rationalize the denominator in 653+2.
Explanation.
To remove radicals in 3+2 with the difference of squares formula, we multiply it with 32.
653+2=653+2(32)(32)=63625352(3)2(2)2=181215+1032=322315+101=322315+10

Subsection 6.5.3 Radical Expressions and Rational Exponents

Example 6.5.5. Radical Expressions and Rational Exponents.

Simplify the expressions using Fact 6.3.2 or Fact 6.3.9.
  1. 1001/2
  2. (64)1/3
  3. 813/4
  4. (127)2/3
Explanation.
  1. 1001/2=(100)=10
  2. (64)1/3=1(64)1/3=1((64)3)=14
  3. 813/4=(814)3=33=27
  4. In this problem the negative can be associated with either the numerator or the denominator, but not both. We choose the numerator.
    (127)2/3=(1273)2=(13273)2=(13)2=(1)2(3)2=19

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. 7z5/9
  2. 54x2/3
  3. (9q5)4/5
  4. y5y24
  5. t3t23
  6. x3
  7. 5(4+a1/2)2
  8. 6(2p5/2)3/5
Explanation.
  1. 7z5/9=7z59
  2. 54x2/3=541x2/3=541x23=54x23
  3. (9q5)4/5=(9)4/5(q5)4/5=(9)4/5q54/5=(95)4q4=(q95)4
  4. y5y24=y5/2y2/4=y5/2+2/4=y10/4+2/4=y12/4=y3
  5. t3t23=t3/2t2/3=t3/22/3=t9/64/6=t5/6=t56
  6. x3=x1/3=(x1/3)1/2=x1/31/2=x1/6=x6
  7. 5(4+a1/2)2=5(4+a1/2)(4+a1/2)=5(42+24a1/2+(a1/2)2)=5(16+8a1/2+a1/22)=5(16+8a1/2+a)=5(16+8a+a)=80+40a+5a
  8. 6(2p5/2)3/5=623/5p5/23/5=623/5p3/2=623/5p3/2=6235p3=685p3

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 t+9=1t for t.
Explanation.
We cannot isolate two radicals, so we will simply square both sides, and later try to isolate the remaining radical.
t+9=1t(t+9)2=(1t)2t+9=1+2t+t after expanding the binomial squared9=1+2t8=2t4=t(4)2=(t)216=t
Because we squared both sides of an equation, we must check the solution by substituting 16 into t+9=1t, and we have:
t+9=1t16+9=?11625=?145=no5
Our solution did not check so there is no solution to this equation. The solution set is the empty set, which can be denoted { } or .

Exercises 6.5.5 Exercises

Square Root and nth Root.

1.
Evaluate the following.
1649=.
2.
Evaluate the following.
25121=.
3.
Evaluate the following.
64=.
4.
Evaluate the following.
81=.

Exercise Group.

5.
Simplify the radical expression or state that it is not a real number.
1255=
6.
Simplify the radical expression or state that it is not a real number.
1444=
7.
Simplify the radical expression or state that it is not a real number.
8=
8.
Simplify the radical expression or state that it is not a real number.
98=

Exercise Group.

9.
Simplify the expression.
37525=
10.
Simplify the expression.
4132121=
11.
Simplify the expression.
5616=
12.
Simplify the expression.
2757=

Exercise Group.

13.
Simplify the expression.
172182=
14.
Simplify the expression.
18111911=
15.
Simplify the expression.
63+252=
16.
Simplify the expression.
252+63=

Exercise Group.

17.
643.
18.
273.
19.
325.
20.
1253.
21.
164.
22.
814.
23.
1286.
24.
1286.
25.
7325.
26.
7273.
27.
135643.
28.
135643.

Rationalizing the Denominator.

29.
Rationalize the denominator and simplify the expression.
4216=
30.
Rationalize the denominator and simplify the expression.
280=
31.
Rationalize the denominator and simplify the expression.
7150=
32.
Rationalize the denominator and simplify the expression.
2125=
33.
Rationalize the denominator and simplify the expression.
92+5=
34.
Rationalize the denominator and simplify the expression.
917+8=
35.
Rationalize the denominator and simplify the expression.
367+8=
36.
Rationalize the denominator and simplify the expression.
2713+5=

Radical Expressions and Rational Exponents.

37.
Without using a calculator, evaluate the expression.
12847=
38.
Without using a calculator, evaluate the expression.
912=
39.
Without using a calculator, evaluate the expression.
(181)34=
40.
Without using a calculator, evaluate the expression.
(19)32=
41.
Without using a calculator, evaluate the expression.
3225=
42.
Without using a calculator, evaluate the expression.
3245=
43.
Without using a calculator, evaluate the expression.
10245=
44.
Without using a calculator, evaluate the expression.
643=

Exercise Group.

45.
Use rational exponents to write the expression.
z9=
46.
Use rational exponents to write the expression.
t6=
47.
Use rational exponents to write the expression.
3r+33=
48.
Use rational exponents to write the expression.
9m+8=

Exercise Group.

49.
Convert the expression to radical notation.
n45 =
50.
Convert the expression to radical notation.
a23 =
51.
Convert the expression to radical notation.
b89 =
52.
Convert the expression to radical notation.
r23 =
53.
Convert the expression to radical notation.
1613x23 =
54.
Convert the expression to radical notation.
516z56 =

Exercise Group.

55.
Simplify the expression, answering with rational exponents and not radicals.
t7t7=
56.
Simplify the expression, answering with rational exponents and not radicals.
r3r3=
57.
Simplify the expression, answering with rational exponents and not radicals.
27m43=
58.
Simplify the expression, answering with rational exponents and not radicals.
32n25=
59.
Simplify the expression, answering with rational exponents and not radicals.
125a3a56=
60.
Simplify the expression, answering with rational exponents and not radicals.
64bb56=
61.
Simplify the expression, answering with rational exponents and not radicals.
cc56=
62.
Simplify the expression, answering with rational exponents and not radicals.
xx310=

Solving Radical Equations.

63.
A pendulum has length L, measured in feet. The time period T that it takes to swing back and forth one time is 4 s. The following formula from physics relates T to L.
T=2πL32
Use this formula to find the length of the pendulum.
64.
A pendulum has length L, measured in feet. The time period T that it takes to swing back and forth one time is 4 s. The following formula from physics relates T to L.
T=2πL32
Use this formula to find the length of the pendulum.
You have attempted 1 of 2 activities on this page.