Section6.5Radical Expressions and Equations Chapter Review
Subsection6.5.1Square and th Root Properties
In Section 1 we defined the square root and th root radicals. When is positive, the expression means a positive number , where times. The square root is just the case where .
When is negative, might not be defined. It depends on whether or not is an even number. When is negative and is odd, is a negative number where times.
There are two helpful properties for simplifying radicals.
List6.5.1.Properties of Radicals for Multiplication and Division
If and are positive real numbers, and 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
Root of a Quotient Property
as long as
Checkpoint6.5.2.
Simplify .
Simplify .
Simplify .
Explanation.
Subsection6.5.2Rationalizing 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.
Example6.5.3.
Rationalize the denominator of the expressions.
Explanation.
First we will simplify .
Now we can rationalize the denominator by multiplying the numerator and denominator by .
Example6.5.4.Rationalize Denominator Using the Difference of Squares Formula.
Rationalize the denominator in .
Explanation.
To remove radicals in with the difference of squares formula, we multiply it with .
Subsection6.5.3Radical Expressions and Rational Exponents
In this problem the negative can be associated with either the numerator or the denominator, but not both. We choose the numerator.
Example6.5.6.More Expressions with Rational Exponents.
Use exponent properties in List 6.3.14 to simplify the expressions, and write all final versions using radicals.
Explanation.
Subsection6.5.4Solving 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.
Example6.5.7.Solving Radical Equations that Require Squaring Twice.
Solve the equation for .
Explanation.
We cannot isolate two radicals, so we will simply square both sides, and later try to isolate the remaining radical.
after expanding the binomial squared
Because we squared both sides of an equation, we must check the solution by substituting into , and we have:
??no
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 .
Exercises6.5.5Exercises
Square Root and th Root.
1.
Evaluate the following.
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2.
Evaluate the following.
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3.
Evaluate the following.
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4.
Evaluate the following.
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Exercise Group.
5.
Simplify the radical expression or state that it is not a real number.
6.
Simplify the radical expression or state that it is not a real number.
7.
Simplify the radical expression or state that it is not a real number.
8.
Simplify the radical expression or state that it is not a real number.
Exercise Group.
9.
Simplify the expression.
10.
Simplify the expression.
11.
Simplify the expression.
12.
Simplify the expression.
Exercise Group.
13.
Simplify the expression.
14.
Simplify the expression.
15.
Simplify the expression.
16.
Simplify the expression.
Exercise Group.
17.
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18.
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20.
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21.
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26.
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27.
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28.
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Rationalizing the Denominator.
29.
Rationalize the denominator and simplify the expression.
30.
Rationalize the denominator and simplify the expression.
31.
Rationalize the denominator and simplify the expression.
32.
Rationalize the denominator and simplify the expression.
33.
Rationalize the denominator and simplify the expression.
34.
Rationalize the denominator and simplify the expression.
35.
Rationalize the denominator and simplify the expression.
36.
Rationalize the denominator and simplify the expression.
Radical Expressions and Rational Exponents.
37.
Without using a calculator, evaluate the expression.
38.
Without using a calculator, evaluate the expression.
39.
Without using a calculator, evaluate the expression.
40.
Without using a calculator, evaluate the expression.
41.
Without using a calculator, evaluate the expression.
42.
Without using a calculator, evaluate the expression.
43.
Without using a calculator, evaluate the expression.
44.
Without using a calculator, evaluate the expression.
Exercise Group.
45.
Use rational exponents to write the expression.
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46.
Use rational exponents to write the expression.
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47.
Use rational exponents to write the expression.
48.
Use rational exponents to write the expression.
Exercise Group.
49.
Convert the expression to radical notation.
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50.
Convert the expression to radical notation.
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51.
Convert the expression to radical notation.
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52.
Convert the expression to radical notation.
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53.
Convert the expression to radical notation.
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54.
Convert the expression to radical notation.
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Exercise Group.
55.
Simplify the expression, answering with rational exponents and not radicals.
56.
Simplify the expression, answering with rational exponents and not radicals.
57.
Simplify the expression, answering with rational exponents and not radicals.
58.
Simplify the expression, answering with rational exponents and not radicals.
59.
Simplify the expression, answering with rational exponents and not radicals.
60.
Simplify the expression, answering with rational exponents and not radicals.
61.
Simplify the expression, answering with rational exponents and not radicals.
62.
Simplify the expression, answering with rational exponents and not radicals.
Solving Radical Equations.
63.
A pendulum has length , measured in feet. The time period that it takes to swing back and forth one time is . The following formula from physics relates to .
Use this formula to find the length of the pendulum.
64.
A pendulum has length , measured in feet. The time period that it takes to swing back and forth one time is . The following formula from physics relates to .
Use this formula to find the length of the pendulum.
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