1. Rational Exponents, Radicals, and Complex Numbers
CHAPTER 10
Rational Exponents, Radicals, and Complex Numbers
10.1 Radical Expressions and Functions
10.2 Rational Exponents
10.3 Multiplying, Dividing, and Simplifying Radicals
10.4 Adding, Subtracting, and Multiplying Radical Expressions
10.5 Rationalizing Numerators and Denominators of Radical Expressions
10.6 Radical Equations and Problem Solving
10.7 Complex Numbers
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2. Radical Expressions and Functions
10.1
Radical Expressions and Functions
1. Find the nth root of a number.
2. Approximate roots using a calculator.
3. Simplify radical expressions.
4. Evaluate radical functions.
5. Find the domain of radical functions.
6. Solve applications involving radical functions.
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3. nth root: The number b is an nth root of a number a if bn = a.
Evaluating nth roots
When evaluating a radical expression , the sign of a and the index n will determine possible outcomes.
If a is nonnegative, then , where and bn = a.
If a is negative and n is even, then there is no real-number root.
If a is negative and n is odd, then , where b is negative and bn = a.
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4. Example Evaluate each square root. a. b. Solution c. Solution Solution
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5. Approximating to two decimal places:
Some roots, like are called irrational because we cannot express their exact value using rational numbers. In fact, writing with the radical sign is the only way we can express its exact value. However, we can approximate using rational numbers.
Approximating to two decimal places:
Approximating to three decimal places:
Note: Remember that the symbol, , means “approximately equal to.”
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6. Example
Approximate the roots using a calculator or table in the endpapers. Round to three decimal places. a. b.
Solution
Solution
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7. Example Find the root. Assume variables represent nonnegative values.c.
Solution
Because (y2)2 = y4.
Solution
Because (6m3)2 = 36m6.
Solution
Because (2x4)5 = 32x20.
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8. Example Find the root. Assume variables represent any real number. a.c.
Solution
Solution
Solution
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9. To find f(3), substitute 3 for x and simplify.
Radical function: A function of the form f(x) =
where P is a polynomial.
Example
Given f(x) = find f(3).
Solution
To find f(3), substitute 3 for x and simplify.
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10. Example Find the domain of each of the following. a.b.
Solution Since the index is even, the radicand must be nonnegative.
Domain:
Solution The radicand must be nonnegative.
Domain:
Conclusion The domain of a radical function with an even index must contain values that keep its radicand nonnegative.
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11. Example If you drop an object, the time (t) it takes in seconds
to fall d feet is given by Find the time it
takes for an object to fall 800 feet.
Understand We are to find the time it takes for an object to fall 800 feet.
Plan Use the formula , replacing d with 800.
Execute
Replace d with 800.
Divide within the radical.
Evaluate the square root.
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12. continued Answer It takes an object 7.071 seconds to fall 800 feet.
Check We can verify the calculations, which we will leave to the viewer.
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13. For which square root is –12.37 the approximation for?b) c) d)
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14. For which square root is –12.37 the approximation for?b) c) d)
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15. Evaluate.
a) 0.2
b) 0.02 c) d)
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16. Evaluate.
a) 0.2
b) 0.02 c) d)
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17. Find the domain of f(x) = .b) c) d)
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18. Find the domain of f(x) = .b) c) d)
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19. 10.2 Rational Exponents 1. Evaluate rational exponents.
2. Write radicals as expressions raised to rational exponents.
3. Simplify expressions with rational number exponents using the rules of exponents.
4. Use rational exponents to simplify radical expressions.
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20. Rational exponents: An exponent that is a fraction.
Rational Exponents with a Numerator of 1
a1/n = where n is a natural number other than 1.
Note: If a is negative and n is odd, then the root is negative.
If a is negative and n is even, then there is no real number root.
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21. Example
Rewrite using radicals, then simplify. a. 491/2 b. 1251/3 c. 641/6
Solution a. b. c.
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22. General Rule for Rational Exponents
where a  0 and m and n are natural numbers other than 1.
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23. Example Rewrite using radicals, then simplify, if possible.
a. 272/3 b. 2433/4 c. 95/2
Solution a. b. c.
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24. Negative Rational Exponents
where a  0, and m and n are natural numbers with n  1.
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25. Example Rewrite using radicals, then simplify, if possible.
a. 251/2 b. 272/3
Solution a. b.
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26. Example Write each of the following in exponential form. a. b.
Solution a. b.
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27. Rules of Exponents Summary
(Assume that no denominators are 0, that a and b are real numbers, and that m and n are integers.)
Zero as an exponent: a0 = 1, where a 0.
00 is indeterminate.
Negative exponents:
Product rule for exponents:
Quotient rule for exponents:
Raising a power to a power:
Raising a product to a power:
Raising a quotient to a power:
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28. Example
Use the rules of exponents to simplify. Write the answer with positive exponents.
Solution
Use the product rule for exponents. (Add the exponents.)
Add the exponents.
Simplify the rational exponent.
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29. Example
Use the rules of exponents to simplify. Write the answer with positive exponents.
Solution
Use the quotient for exponents. (Subtract the exponents.)
Rewrite the subtraction as addition.
Add the exponents.
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30. Example
Perform the indicated operations. Write the result using a radical. a. b.
Solution a. b.
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31. Simplify. a) b) c) d)
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32. Simplify. a) b) c) d)
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33. Simplify.
a) 5
b) 25
c) 25
d) 5
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34. Simplify.
a) 5
b) 25
c) 25
d) 5
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35. Simplify.
a) 4 b)
c) 4 d)
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36. Simplify.
a) 4 b)
c) 4 d)
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