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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.

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=

Exercise Group.

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.
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