0-4 nth Roots and Real Exponents
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1 Evaluate Because there is no real number that can be squared to produce 100, is not a real number. not a real number 3. esolutions Manual - Powered by Cognero Page 1
2 4. 5. Because there is no real number that can be raised to the fourth power to produce 81, is not a real number. not a real number esolutions Manual - Powered by Cognero Page 2
3 Simplify. Because the index is odd, it is not necessary to use absolute value. 10. Because you are taking an even root of an even power and the result is an odd power, you must use the absolute value of a 5. 2 a You are taking an even root of an even power, but the result is an even power, so you do not need to use the absolute value of y 2. esolutions Manual - Powered by Cognero Page 3
4 12. Because the index is odd, it is not necessary to use absolute value. 13. While you are taking an even root of an even power, the result is an even power, so you do not need to use the absolute value of y Because you are taking an even root of an even power and the result is an odd power, you must use the absolute value of (z 2) 7. esolutions Manual - Powered by Cognero Page 4
5 15. Because you are taking an even root of an even power and the result is an odd power, you must use the absolute value of a Because the index is odd, it is not necessary to use absolute value. esolutions Manual - Powered by Cognero Page 5
6 17. Simplify. Because you are taking an even root of an odd power, it is not necessary to use absolute value. Therefore, simplifies to or. or 18. or esolutions Manual - Powered by Cognero Page 6
7 esolutions Manual - Powered by Cognero Page 7
8 esolutions Manual - Powered by Cognero Page 8
9 We can further simplify this expression by rationalizing the denominator using rational exponents. esolutions Manual - Powered by Cognero Page 9
10 23. BOATING The motion comfort ratio M of a boat is given by M =, where D is the water displacement of the boat in pounds, B is the boat s beam or width in feet, W is the boat s length in feet at the waterline, and A is the boat s overall length in feet. The higher the ratio, the greater the level of comfort experienced by those on board as the boat encounters waves. a. Find the motion comfort ratio of the boat shown below. b. Find the beam of a boat to the nearest foot with a comfort ratio of 27 that displaces 15,000 pounds of water, has a waterline length of 30.4 feet, and an overall length of 32.3 feet. a. From the information given, we know that D = 13,800, B = 10, W = 26.2, and A = Use a calculator to evaluate the given formula relating these variables to find M, the motion comfort ratio of the boat. b. From the information give, we know that M = 27, D = 15,000, W = 30.4, and A = Use a calculator to solve the given formula for B, the beam of the boat. Move everything attached to the B from the right side of the equation to the left side. Do this by multiplying everything in the denominator and dividing everything in the denominator. esolutions Manual - Powered by Cognero Page 10
11 The beam of a boat is about 12.0 ft. a b ft 24. CARS The value of a car depreciates or declines over the course of its useful life. The new value V and the original value v of a car are related by the formula V = v(1 r) n, where r is the rate of depreciation per year and n is the number of years. Suppose the current value of a used car is $12,000. What would be the value of the car after 18 months at an annual depreciation rate of 20%? From the information given, we know that v = 12,000 and r = 0.2. The time is given in months but the depreciation formula requires that the time be in years. Since 18 months is equivalent to 1.5 years, n = 1.5. Evaluate the given equation to find the value of the car V. The value of the car would be $ $8, esolutions Manual - Powered by Cognero Page 11
12 25. Evaluate esolutions Manual - Powered by Cognero Page 12
13 29. MUSIC The note progression of the twelve tone scale is comprised of a series of half tones. In order for an instrument to be in tune, the frequency of each note has an optimum ratio with the frequency of middle C, called the perfect 1st. The optimum frequency ratio r, expressed as a decimal, can be calculated using, where n is the number of half tones the note is above the perfect 1st, including the note itself. a. Approximate the optimum frequency ratio of the middle 3rd with the perfect 1st. b. Without the use of a calculator, approximate the optimum frequency ratio of the perfect 8th and the perfect 1st. Justify your answer. a. The middle 3rd is 4 half tones above the perfect 1st, so evaluate for n = 4. The optimum frequency ratio of the middle 3rd with the perfect 1st is b. The perfect 8th is 12 half tones above the perfect 1st. Evaluating for n = 12, yields, which simplifies to 2 1 or 2. a. about 1.26 b. 2; sample answer: The perfect 8th is 12 half tones above the perfect 1st. Evaluating for n = 12, yields, which simplifies to 2 1 or 2. esolutions Manual - Powered by Cognero Page 13
14 30. Simplify. Because the index is odd, it is not necessary to use absolute value. 31. Because the index is odd, it is not necessary to use absolute value. 32. Because you are taking an even root of an even power and the result is an odd power, you must use the absolute value of b and c 5. esolutions Manual - Powered by Cognero Page 14
15 33. Because you are taking an even root of an even power and the result is an odd power, you must use the absolute value of y and z 3. It is not necessary to use absolute value on x because is an even root of an odd power. 34. Because you are taking an even root of an even power and the result is an odd power, you must use the absolute value of c. You do not need an absolute value of d because the term guarantees that y is nonnegative. 35. Because the index is odd, it is not necessary to use absolute value. esolutions Manual - Powered by Cognero Page 15
Chapter Review. Write each expression using exponents SOLUTION: The base 6 is a factor 5 times. So, the exponent is 5.
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Simplify. 1. i 10 2. i 2 + i 8 3. i 3 + i 20 4. i 100 5. i 77 esolutions Manual - Powered by Cognero Page 1 6. i 4 + i 12 7. i 5 + i 9 8. i 18 Simplify. 9. (3 + 2i) + ( 4 + 6i) 10. (7 4i) + (2 3i) 11.
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Simplify. 1. i 10 1 2. i 2 + i 8 0 3. i 3 + i 20 1 i esolutions Manual - Powered by Cognero Page 1 4. i 100 1 5. i 77 i 6. i 4 + i 12 2 7. i 5 + i 9 2i esolutions Manual - Powered by Cognero Page 2 8.
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