6.7E: Exercises
- Page ID
- 30546
Practice Makes Perfect
Use the Definition of a Negative Exponent
In the following exercises, simplify.
- \(4^{−2}\)
- \(10^{−3}\)
- \(3^{−4}\)
- \(10^{−2}\)
- Answer
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- \(\frac{1}{81}\)
- \(\frac{1}{100}\)
- \(5^{−3}\)
- \(10^{−5}\)
- \(2^{−8}\)
- \(10^{−2}\)
- Answer
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- \(\frac{1}{256}\)
- \(\frac{1}{100}\)
- \(\dfrac{1}{c^{−5}}\)
- \(\dfrac{1}{3^{−2}}\)
- \(\dfrac{1}{c^{−5}}\)
- \(\dfrac{1}{5^{−2}}\)
- Answer
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- \(c^5\)
- 25
- \(\dfrac{1}{q^{−10}}\)
- \(\dfrac{1}{10^{−3}}\)
- \(\dfrac{1}{t^{−9}}\)
- \(\dfrac{1}{10^{−4}}\)
- Answer
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- \(t^9\)
- \(10000\)
- \(\left(\dfrac{5}{8}\right)^{−2}\)
- \(\left(−\dfrac{3m}{n}\right)^{−2}\)
- \(\left(\dfrac{3}{10}\right)^{−2}\)
- \(\left(−\dfrac{2c}{d}\right)^{−3}\)
- Answer
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- \(\dfrac{100}{9}\)
- \(−\dfrac{d^3}{8c^3}\)
- \(\left(\dfrac{4}{9}\right)^{−3}\)
- \(\left(−\dfrac{u^2}{2v}\right)^{−5}\)
- \(\left(\dfrac{7}{2}\right)^{−3}\)
- \(\left(−\dfrac{3}{xy^2}\right)^{−3}\)
- Answer
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- \(\dfrac{8}{343}\)
- \(−\dfrac{x^{3}y^6}{27}\)
- \((−5)^{−2}\)
- \(−5^{−2}\)
- \(\left(−\frac{1}{5}\right)^{−2}\)
- \(−\left(\frac{1}{5}\right)^{−2}\)
- \((−7)^{−2}\)
- \(−7^{−2}\)
- \(\left(−\frac{1}{7}\right)^{−2}\)
- \(−\left(\frac{1}{7}\right)^{−2}\)
- Answer
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- \(\frac{1}{49}\)
- \(−\frac{1}{49}\)
- 49
- −49
- \(−3^{−3}\)
- \(\left(−\frac{1}{3}\right)^{−3}\)
- \(−\left(\frac{1}{3}\right)^{−3}\)
- \((−3)^{−3}\)
- \(−5^{−3}\)
- \(\left(−\frac{1}{5}\right)^{−3}\)
- \(−\left(\frac{1}{5}\right)^{−3}\)
- \((−5)^{−3}\)
- Answer
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- \(−\frac{1}{125}\)
- −125
- −125
- \(−\frac{1}{125}\)
- \(3·5^{−1}\)
- \((3·5)^{−1}\)
- \(2·5^{−1}\)
- \((2·5)^{−1}\)
- Answer
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- \(\frac{2}{5}\)
- \(\frac{1}{10}\)
- \(4·5^{−2}\)
- \((4·5)^{−2}\)
- \(3·4^{−2}\)
- \((3·4)^{−2}\)
- Answer
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- \(\frac{3}{16}\)
- \(\frac{1}{144}\)
- \(m^{−4}\)
- \((x^3)^{−4}\)
- \(b^{−5}\)
- \((k^2)^{−5}\)
- Answer
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- \(\dfrac{1}{b^5}\)
- \(\dfrac{1}{k^{10}}\)
- \(p^{−10}\)
- \((q^6)^{−8}\)
- \(s^{−8}\)
- \((a^9)^{−10}\)
- Answer
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- \(\dfrac{1}{s^8}\)
- \(\dfrac{1}{a^{90}}\)
- \(7n^{−1}\)
- \((7n)^{−1}\)
- \((−7n)^{−1}\)
- \(6r^{−1}\)
- \((6r)^{−1}\)
- \((−6r)^{−1}\)
- Answer
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- \(\dfrac{6}{r}\)
- \(\dfrac{1}{6r}\)
- \(−\dfrac{1}{6r}\)
- \((3p)^{−2}\)
- \(3p^{−2}\)
- \(−3p^{−2}\)
- \((2q)^{−4}\)
- \(2q^{−4}\)
- \(−2q^{−4}\)
- Answer
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- \(\dfrac{1}{16q^4}\)
- \(\dfrac{2}{q^4}\)
- \(−\dfrac{2}{q^4}\)
Simplify Expressions with Integer Exponents
In the following exercises, simplify.
- \(b^{4}b^{−8}\)
- \(r^{−2}r^5\)
- \(x^{−7}x^{−3}\)
- \(s^3·s^{−7}\)
- \(q^{−8}·q^3\)
- \(y^{−2}·y^{−5}\)
- Answer
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- \(\dfrac{1}{s^4}\)
- \(\dfrac{1}{q^5}\)
- \(\dfrac{1}{y^7}\)
- \(a^3·a^{−3}\)
- \(a·a^3\)
- \(a·a^{−3}\)
- \(y^5·y^{−5}\)
- \(y·y^5\)
- \(y·y^{−5}\)
- Answer
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- 1
- \(y^6\)
- \(\dfrac{1}{y^4}\)
\(p^5·p^{−2}·p^{−4}\)
\(x^4·x^{−2}·x^{−3}\)
- Answer
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\(\dfrac{1}{x}\)
\((w^{4}x^{−5})(w^{−2}x^{−4})\)
\((m^{3}n^{−3})(m^{−5}n^{−1})\)
- Answer
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\(\dfrac{1}{m^{2}n^4}\)
\((uv^{−2})(u^{−5}v^{−3})\)
\((pq^{−4})(p^{−6}q^{−3})\)
- Answer
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\(\dfrac{1}{p^{5}q^{7}}\)
\((−6c^{−3}d^9)(2c^{4}d^{−5})\)
\((−2j^{−5}k^8)(7j^{2}k^{−3})\)
- Answer
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\(−\dfrac{14k^5}{j^3}\)
\((−4r^{−2}s^{−8})(9r^{4}s^3)\)
\((−5m^{4}n^6)(8m^{−5}n^{−3})\)
- Answer
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\(−\dfrac{40n^3}{m}\)
\((5x^2)^{−2}\)
\((4y^3)^{−3}\)
- Answer
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\(\dfrac{1}{64y^9}\)
\((3z^{−3})^2\)
\((2p^{−5})^2\)
- Answer
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\(\dfrac{4}{p^{10}}\)
\(\dfrac{t^{9}}{t^{−3}}\)
\(\dfrac{n^{5}}{n^{−2}}\)
- Answer
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\(n^7\)
\(\dfrac{x^{−7}}{x^{−3}}\)
\(\dfrac{y^{−5}}{y^{−10}}\)
- Answer
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\(y^5\)
Convert from Decimal Notation to Scientific Notation
In the following exercises, write each number in scientific notation.
57,000
340,000
- Answer
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\(3.4 \times 10^{5}\)
8,750,000
1,290,000
- Answer
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\(1.29 \times 10^{6}\)
0.026
0.041
- Answer
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\(4.1 \times 10^{-2}\)
0.00000871
0.00000103
- Answer
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\(1.03 \times 10^{-6}\)
Convert Scientific Notation to Decimal Form
In the following exercises, convert each number to decimal form.
\(5.2 \times 10^{2}\)
\(8.3 \times 10^{2}\)
- Answer
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830
\(7.5 \times 10^{6}\)
\(1.6 \times 10^{10}\)
- Answer
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16,000,000,000
\(2.5 \times 10^{-2}\)
\(3.8 \times 10^{-2}\)
- Answer
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0.038
\(4.13 \times 10^{-5}\)
\(1.93 \times 10^{-5}\)
- Answer
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0.0000193
Multiply and Divide Using Scientific Notation
In the following exercises, multiply. Write your answer in decimal form.
\(\left(3 \times 10^{-5}\right)\left(3 \times 10^{9}\right)\)
\(\left(2 \times 10^{2}\right)\left(1 \times 10^{-4}\right)\)
- Answer
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0.02
\(\left(7.1 \times 10^{-2}\right)\left(2.4 \times 10^{-4}\right)\)
\(\left(3.5 \times 10^{-4}\right)\left(1.6 \times 10^{-2}\right)\)
- Answer
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\(5.6 \times 10^{-6}\)
In the following exercises, divide. Write your answer in decimal form.
\(\dfrac{7 \times 10^{-3}}{1 \times 10^{-7}}\)
\(\dfrac{5 \times 10^{-2}}{1 \times 10^{-10}}\)
- Answer
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500,000,000
\(\dfrac{6 \times 10^{4}}{3 \times 10^{-2}}\)
\(\dfrac{8 \times 10^{6}}{4 \times 10^{-1}}\)
- Answer
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20,000,000
Everyday Math
The population of the United States on July 4, 2010 was almost 310,000,000. Write the number in scientific notation.
The population of the world on July 4, 2010 was more than 6,850,000,000. Write the number in scientific notation
- Answer
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\(6.85 \times 10^{9}\)
The average width of a human hair is 0.0018 centimeters. Write the number in scientific notation.
The probability of winning the 2010 Megamillions lottery was about 0.0000000057. Write the number in scientific notation.
- Answer
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\(5.7 \times 10^{-9}\)
In \(2010,\) the number of Facebook users each day who changed their status to 'engaged" was \(2 \times 10^{4} .\) Convert this number
to decimal form.
At the start of \(2012,\) the US federal budget had a deficit of more than \(\$ 1.5 \times 10^{13} .\) Convert this number to decimal form.
- Answer
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15,000,000,000,000
The concentration of carbon dioxide in the atmosphere is \(3.9 \times 10^{-4} .\) Convert this number to decimal form.
The width of a proton is \(1 \times 10^{-5}\) of the width of an atom. Convert this number to decimal form.
- Answer
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0.00001
Health care costs The Centers for Medicare and Medicaid projects that consumers will spend more than $4 trillion on health care by 2017.
- Write 4 trillion in decimal notation.
- Write 4 trillion in scientific notation.
Coin production In 1942, the U.S. Mint produced 154,500,000 nickels. Write 154,500,000 in scientific notation.
- Answer
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\(1.545 \times 10^{8}\)
Distance The distance between Earth and one of the brightest stars in the night star is 33.7 light years. One light year is about 6,000,000,000,000 (6 trillion), miles.
- Write the number of miles in one light year in scientific notation.
- Use scientific notation to find the distance between Earth and the star in miles. Write the answer in scientific notation.
Debt At the end of fiscal year 2015 the gross United States federal government debt was estimated to be approximately $18,600,000,000,000 ($18.6 trillion), according to the Federal Budget. The population of the United States was approximately 300,000,000 people at the end of fiscal year 2015.
- Write the debt in scientific notation.
- Write the population in scientific notation.
- Find the amount of debt per person by using scientific notation to divide the debt by the population. Write the answer in scientific notation.
- Answer
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- \(1.86 \times 10^{13}\)
- \(3 \times 10^{8}\)
- \(6.2 \times 10^{4}\)
Writing Exercises
- Explain the meaning of the exponent in the expression \(2^{3}\).
- Explain the meaning of the exponent in the expression \(2^{-3}\).
When you convert a number from decimal notation to scientific notation, how do you know if the exponent will be positive or negative?
- Answer
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answers will vary
Self Check
a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
b. Overall, after looking at the checklist, do you think you are well-prepared for the next section? Why or why not?
More Practice with Exponents in Quotients
Quotients
Exercise \(\PageIndex{89}\)
\( \bigstar \) Simplify. (Assume all variables represent nonzero numbers.)
|
1. \( \displaystyle \frac { 10 ^ { 2 } \cdot 10 ^ { 4 } } { 10 ^ { 5 } } \\[6pt]\) 2. \( \displaystyle \frac { 7 ^ { 5 } \cdot 7 ^ { 9 } } { 7 ^ { 2 } } \\[6pt]\) 3. \( \displaystyle \frac { a ^ { 8 } \cdot a ^ { 6 } } { a ^ { 5 } } \\[6pt]\) 4. \( \displaystyle \frac { b ^ { 4 } \cdot b ^ { 10 } } { b ^ { 8 } } \\[6pt]\) |
5. \( \displaystyle\frac { a ^ { 8 } \cdot a ^ { - 3 } } { a ^ { - 6 } } \\[6pt]\) 6. \( \displaystyle\frac { b ^ { - 10 } \cdot b ^ { 4 } } { b ^ { - 2 } } \\[6pt]\) 7. \( \displaystyle\frac { 25 x ^ { - 3 } y ^ { 2 } } { 5 x ^ { - 1 } y ^ { - 3 } } \\[6pt]\) 8. \( \displaystyle\frac { - 9 x ^ { - 1 } y ^ { 3 } z ^ { - 5 } } { 3 x ^ { - 2 } y ^ { 2 } z ^ { - 1 } } \\[6pt]\) |
9. \( \displaystyle \frac { 40 x ^ { 5 } y ^ { 3 } z } { 4 x ^ { 2 } y ^ { 2 } z } \\[6pt]\) 10. \( \displaystyle \frac { 8 x ^ { 2 } y ^ { 5 } z ^ { 3 } } { 16 x ^ { 2 } y z } \\[6pt]\) 11. \( \displaystyle \frac { 24 a ^ { 8 } b ^ { 3 } ( a - 5 b ) ^ { 10 } } { 8 a ^ { 5 } b ^ { 3 } ( a - 5 b ) ^ { 2 } } \\[6pt]\) 12. \( \displaystyle \frac { 175 m ^ { 9 } n ^ { 5 } ( m + n ) ^ { 7 } } { 25 m ^ { 8 } n ( m + n ) ^ { 3 } } \\[6pt]\) |
- Answers to odd exercises.
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1. \(10 \\[6pt]\)
3. \(a^{9} \)
5. \(a^{11} \\[6pt]\)
7. \( \displaystyle\frac { 5 y ^ { 5 } } { x ^ { 2 } } \)
9. \(10x^{3}y \\[6pt]\)
11. \(3a^{3}(a − 5b)^{8} \)
Powers of Quotients
Exercise \(\PageIndex{90} \)
\( \bigstar \) Simplify. (Assume all variables represent nonzero numbers.)
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1. \(\left( \displaystyle \frac { - 3 a b ^ { 2 } } { 2 c ^ { 3 } } \right) ^ {3 } \\[6pt]\) 2. \(\left( \displaystyle \frac { - 10 a ^ { 3 } b } { 3 c ^ { 2 } } \right) ^ {2 } \\[6pt]\) 3. \(\left( \displaystyle \frac { - 2 x y ^ { 4 } } { z ^ { 3 } } \right) ^ {4 } \\[6pt]\) 4. \(\left( \displaystyle \frac { - 7 x ^ { 9 } y } { z ^ { 4 } } \right) ^ {3 } \) |
5. \(\left( \displaystyle \frac { 12 x ^ { 3 } y ^ { 2 } z } { 2 x ^ { 7 } y z ^ { 8 } } \right) ^ {3 } \\[6pt]\) 6. \(\left( \displaystyle \frac { 150 x y ^ { 8 } z ^ { 2 } } { 90 x ^ { 7 } y ^ { 2 } z } \right) ^ {2 } \\[6pt]\) 7. \(\left( \displaystyle \frac { 2 x ^ { - 3 } z } { y ^ { 2 } } \right) ^ {- 5 } \\[6pt]\) 8. \(\left( \displaystyle \frac { 5 x ^ { 5 } z ^ { - 2 } } { 2 y ^ { - 3 } } \right) ^ {- 3 } \) |
9. \(\left( \displaystyle \frac { x y ^ { 2 } } { z ^ { 3 } } \right) ^ {3 } \\[6pt]\) 10. \(\left( \displaystyle \frac { 2 x ^ { 2 } y ^ { 3 } } { z } \right) ^ {4 } \\[6pt]\) 11. \(\left( \displaystyle \frac { - 9 a ^ { - 3 } b ^ { 4 } c ^ { - 2 } } { 3 a ^ { 3 } b ^ { 5 } c ^ { - 7 } } \right) ^ {- 4 } \\[6pt]\) 12. \(\left( \displaystyle \frac { - 15 a ^ { 7 } b ^ { 5 } c ^ { - 8 } } { 3 a ^ { - 6 } b ^ { 2 } c ^ { 3 } } \right) ^ {- 3 } \) |
- Answers to odd exercises:
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1. \(- \displaystyle\frac { 27 a ^ { 3 } b ^ { 6 } } { 8 c ^ { 9 } } \\[6pt]\)
3. \( \displaystyle\frac { 16 x ^ { 4 } y ^ { 16 } } { z ^ { 12 } } \)
5. \( \displaystyle\frac { 216 y ^ { 3 } } { x ^ { 12 } z ^ { 21 } } \\[6pt]\)
7. \( \displaystyle\frac { x ^ { 15 } y ^ { 10 } } { 32 z ^ { 5 } } \)
9. \( \displaystyle\frac { x ^ { 3 } y ^ { 6} } { z ^ { 9 } } \\[6pt]\)
11. \( \displaystyle\frac { a ^ { 24 } b ^ { 4 } } { 81 c ^ { 20 } } \)