In the following exercises, simplify each expression using the properties for exponents.
3. ⓐ \(n^{19}·n^{12}\) ⓑ \(3^x·3^6\) ⓒ \(7w^5·8w\) ⓓ \(a^4·a^3·a^9\)
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ⓐ \(n^{31}\) ⓑ \(3^{x+6}\) ⓒ \(56w^6\)
ⓓ \(a^{16}\)
4. ⓐ \(q^{27}·q^{15}\) ⓑ \(5^x·5^{4x}\) ⓒ \(9u^{41}·7u^{53}\)
ⓓ \(c^5·c^{11}·c^2\)
5. \(m^x·m^3\)
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\(m^{x+3}\)
7. \(y^a·y^b\)
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\(y^{a+b}\)
9. ⓐ \(\dfrac{x^{18}}{x^3}\) ⓑ \(\dfrac{5^{12}}{5^3}\) ⓒ \(\dfrac{q^{18}}{q^{36}}\) ⓓ \(\dfrac{10^2}{10^3}\)
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ⓐ \(x^{15}\) ⓑ \(5^9\) ⓒ \(\dfrac{1}{q^{18}}\) ⓓ \(\dfrac{1}{10}\)
10. ⓐ \(\dfrac{y^{20}}{y^{10}}\) ⓑ \(\dfrac{7^{16}}{7^2}\) ⓒ \(\dfrac{t^{10}}{t^{40}}\) ⓓ \(\dfrac{8^3}{8^5}\)
11. ⓐ \(\dfrac{p^{21}}{p^7}\) ⓑ \(\dfrac{4^{16}}{4^4}\) ⓒ \(\dfrac{b}{b^9}\) ⓓ \(\dfrac{4}{4^6}\)
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ⓐ \(p^{14}\) ⓑ \(4^{12}\) ⓒ \(\dfrac{1}{b^8}\) ⓓ \(\dfrac{1}{4^5}\)
12. ⓐ \(\dfrac{u^{24}}{u^3}\) ⓑ \(\dfrac{9^{15}}{9^5}\) ⓒ \(\dfrac{x}{x^7}\) ⓓ \(\dfrac{10}{10^3}\)
13. ⓐ \(20^0\) ⓑ \(b^0\)
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ⓐ 1 ⓑ 1
15. ⓐ \(−27^0\) ⓑ \(−(27^0)\)
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ⓐ \(−1\) ⓑ \(−1\)
16. ⓐ \(−15^0\) ⓑ \(−(15^0)\)
Use the Definition of a Negative Exponent
In the following exercises, simplify each expression.
17. ⓐ \(a^{−2}\) ⓑ \(10^{−3}\) ⓒ \(\dfrac{1}{c^{−5}}\) ⓓ \(\dfrac{1}{3^{−2}}\)
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ⓐ \(\dfrac{1}{a^{2}}\) ⓑ \(\dfrac{1}{1000}\) ⓒ \(c^{5}\) ⓓ \(9\)
18. ⓐ \(b^{−4}\) ⓑ \(10^{−2}\) ⓒ \(\dfrac{1}{c^{−5}}\) ⓓ \(\dfrac{1}{5^{−2}}\)
19. ⓐ \(r^{−3}\) ⓑ \(10^{−5}\) ⓒ \(\dfrac{1}{q^{−10}}\) ⓓ \(\dfrac{1}{10^{−3}}\)
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ⓐ \(\dfrac{1}{r3}\) ⓑ \(\dfrac{1}{100,000}\) ⓒ \(q^{10}\) ⓓ \(1,000\)
20. ⓐ \(s^{−8}\) ⓑ \(10^{−2}\) ⓒ \(\dfrac{1}{t^{−9}}\) ⓓ \(\dfrac{1}{10^{−4}}\)
21. ⓐ \(\left(\dfrac{5}{8}\right)^{-2}\) ⓑ \(\left(−\dfrac{b}{a}\right)^{−2}\)
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ⓐ \(\dfrac{64}{25}\) ⓑ \(\dfrac{a^{2}}{b^{2}}\)
22. ⓐ \(\left(\dfrac{3}{10}\right)^{−2}\) ⓑ \(\left(−\dfrac{2}{z}\right)^{−3}\)
23. ⓐ \(\left(\dfrac{4}{9}\right)^{−3}\) ⓑ \(\left(−\dfrac{u}{v}\right)^{−5}\)
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ⓐ \(\dfrac{729}{64}\) ⓑ \(−\dfrac{v^{5}}{u^{5}}\)
24. ⓐ \(\left(\dfrac{7}{2}\right)^{−3}\) ⓑ \(\left(−\dfrac{3}{x}\right)^{−3}\)
25. ⓐ \((−5)^{−2}\) ⓑ \(−5^{−2}\) ⓒ \(\left(−\dfrac{1}{5}\right)^{−2}\) ⓓ \(−\left(\dfrac{1}{5}\right)^{−2}\)
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ⓐ \(\dfrac{1}{25}\) ⓑ \(−\dfrac{1}{25}\) ⓒ \(25\) ⓓ \(−25\)
26. ⓐ \(−5^{−3}\) ⓑ \(\left(−\dfrac{1}{5}\right)^{−3}\) ⓒ \(−\left(\dfrac{1}{5}\right)^{−3}\) ⓓ \((−5)^{−3}\)
27. ⓐ \(3·5^{−1}\) ⓑ \((3·5)^{−1}\)
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ⓐ \(\dfrac{3}{5}\) ⓑ \(\dfrac{1}{15}\)
28. ⓐ \(3·4^{−2}\) ⓑ \((3·4)^{−2}\)
In the following exercises, simplify each expression using the Product Property.
29. ⓐ \(b^{4}b^{−8}\) ⓑ \((w^{4}x^{−5})(w^{−2}x^{−4})\)) ⓒ \((−6c^{−3}d^9)(2c^4d^{−5})\)
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ⓐ \(\dfrac{1}{b^{4}}\) ⓑ \(\dfrac{w^{2}}{x^{9}}\) ⓒ \(−12cd^{4}\)
30. ⓐ \(s^{3}·s^{−7}\) ⓑ \((m^{3}n^{−3})(m^{5}n^{−1})\)
ⓒ \((−2j^{−5}k^{8})(7j^{2}k^{−3})\)
31. ⓐ \(a^{3}·a^{−3}\) ⓑ \((uv^{−2})(u^{−5}v^{−3})\)
ⓒ \((−4r^{−2}s^{−8})(9r^{4}s^{3})\)
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ⓐ \(1\) ⓑ \(\dfrac{1}{u^{4}v^{5}}\) ⓒ \(−36\dfrac{r^{2}}{j^{5}}\)
32. ⓐ \(y^{5}·y^{−5}\) ⓑ \((pq^{−4})(p^{−6}q^{−3})\)
ⓒ \((−5m^{4}n^{6})(8m^{−5}n^{−3})\)
33. \(p^{5}·p^{−2}·p^{−4}\)
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\(\dfrac{1}{p}\)
34. \(x^{4}·x^{−2}·x^{−3}\)
In the following exercises, simplify each expression using the Power Property.
35. ⓐ \((m^4)^2\) ⓑ \((10^3)^6\) ⓒ \((x^3)^{−4}\)
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ⓐ \(m^{8}\) ⓑ \(10^{18}\) ⓒ \(\dfrac{1}{x^{12}}\)
36. ⓐ \((b^{2})^{7}\) ⓑ \((3^8)^2\) ⓒ \((k^2)^{−5}\)
37. ⓐ \((y^3)^x\) ⓑ \((5^x)^x\) ⓒ \((q^6)^{−8}\)
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ⓐ \(y^{3x}\) ⓑ \(5^{xy}\) ⓒ \(\dfrac{1}{q^{48}}\)
38. ⓐ \((x^2)^y\) ⓑ \((7^a)^b\) ⓒ \((a^9)^{−10}\)
In the following exercises, simplify each expression using the Product to a Power Property.
39. ⓐ \((−3xy)^2\) ⓑ \((6a)^0\) ⓒ \((5x^2)^{−2}\) ⓓ \((−4y^{−3})^2\)
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ⓐ \(9x^2y^2\) ⓑ 1 ⓒ \(\dfrac{1}{25x^4}\) ⓓ \(\dfrac{16}{y^6}\)
40. ⓐ \((−4ab)^2\) ⓑ \((5x)^0\) ⓒ \((4y^3)^{−3}\) ⓓ \((−7y^{−3})^2\)
41. ⓐ \((−5ab)^3\) ⓑ \((−4pq)^0\) ⓒ \((−6x^3)^{−2}\) ⓓ \((3y^{−4})^2\)
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ⓐ \(−125a^3b^3\) ⓑ 1 ⓒ \(\dfrac{1}{36x^6}\) ⓓ \(\dfrac{9}{y^8}\)
42. ⓐ \((−3xyz)^4\) ⓑ \((−7mn)^0\) ⓒ \((−3x^3)^{−2}\)
ⓓ \((2y^{−5})^2\)
In the following exercises, simplify each expression using the Quotient to a Power Property.
43. ⓐ \((p^2)^5\) ⓑ \(\left(\dfrac{x}{y}\right)^{−6}\) ⓒ \(\left(\dfrac{2xy^2}{z}\right)^3\) ⓓ \(\left(\dfrac{4p^{−3}}{q^2}\right)^2\)
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ⓐ \(\dfrac{p^5}{32}\) ⓑ \(\dfrac{y^6}{x^6}\) ⓒ \(\dfrac{8x^3y^6}{z^3}\)
ⓓ \(\dfrac{16}{p^6q^4}\)
44. ⓐ \(\left(\dfrac{x}{3}\right)^4\) ⓑ \(\left(\dfrac{a}{b}\right)^{−5}\) ⓒ \(\left(\dfrac{2xy^2}{z}\right)^3\) ⓓ \(\left(\dfrac{x^3y}{z^4}\right)^2\)
45. ⓐ \(\left(\dfrac{a}{3b}\right)^4\) ⓑ \(\left(\dfrac{5}{4m}\right)^{−2}\) ⓒ \(\left(\dfrac{3a^{−2}b^3}{c^3}\right)^{−2}\) ⓓ \(\left(\dfrac{p^{−1}q^4}{r^{−4}}\right)^2\)
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ⓐ \(\dfrac{a^4}{81b^4}\) ⓑ \(\dfrac{16m^2}{25}\) ⓒ \(\dfrac{a^4c^4}{9b^6}\) ⓓ \(\dfrac{q^8r^8}{p^2}\)
46. ⓐ \(\left(\dfrac{x^2}{y}\right)^3\) ⓑ \(\left(\dfrac{10}{3q}\right)^{−4}\) ⓒ \(\left(\dfrac{2x^3y^4}{3z^2}\right)^5\) ⓓ \(\left(\dfrac{5a^3b^{−1}}{2c^4}\right)^{−3}\)
In the following exercises, simplify each expression by applying several properties.
47. ⓐ \((5t^2)^3(3t)^2\) ⓑ \(\dfrac{(t^2)^5(t^{−4})^2}{(t^3)^7}\) ⓒ \(\left(\dfrac{2xy^2}{x^3y^{−2}}\right)^2\left(\dfrac{12xy^3}{x^3y^{−1}}\right)^{−1}\)
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ⓐ \(1125t^8\) ⓑ \(\dfrac{1}{t^{19}}\) ⓒ \(\dfrac{y^4}{3x^2}\)
48. ⓐ \((10k^4)^3(5k^6)^2\) ⓑ \(\dfrac{(q^3)^6(q^{−2})^3}{(q^4)^8}\)
49. ⓐ \((m^2n)^2(2mn^5)^4\) ⓑ \(\dfrac{(−2p^{−2})^4(3p^4)^2}{(−6p^3)^2}\)
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ⓐ \(16m^8n^{22}\) ⓑ \(\dfrac{4}{p^6}\)
50. ⓐ \((3pq^4)^2(6p^6q)^2\) ⓑ \(\dfrac{(−2k^{−3})^2(6k^2)^4}{(9k^4)^2}\)
Mixed Practice
In the following exercises, simplify each expression.
51. ⓐ \(7n^{−1}\) ⓑ \((7n)^{−1}\) ⓒ \((−7n)^{−1}\)
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ⓐ \(\dfrac{7}{n}\) ⓑ \(\dfrac{1}{7n}\) ⓒ \(−\dfrac{1}{7n}\)
52. ⓐ \(6r^{−1}\) ⓑ \((6r)^{−1}\) ⓒ \((−6r)^{−1}\)
In the following exercises, write each number in scientific notation.
In the following exercises, convert each number to decimal form.
In the following exercises, multiply or divide as indicated. Write your answer in decimal form.