Thornton Bartlett
08/27/2023 · Middle School
\( \begin{array}{ll}\text { 62. }\left(\frac{15 a^{2} b^{-2}}{-3 a b^{-3}}\right)^{-2} & \text { 63. }\left(x^{-3} y^{-1}\right)^{-1}\left(x^{-3} y^{0}\right)^{2} \\ \text { 65. }\left[\frac{\left(a^{3} b^{5}\right)^{2}}{a^{5} b^{2}}\right]^{-1} & \text { 66. }\left(\frac{s^{-3}}{4 t}\right)^{-3}\left(\frac{5 t}{s^{-7}}\right)^{-2} \\ \text { 68. }\left[\left(\frac{x^{5} y^{2}}{x^{-3} y}\right)^{-2}\left(\frac{y^{-3}}{2 x^{5}}\right)^{3}\right]^{-1} & \text { 69. }\left[\frac{\left(a^{-5} b^{2}\right)^{-1}}{\left(-a^{1} b^{4} c^{-1}\right)^{2}}\right]^{-3}\end{array} \)
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
Let's simplify each expression step by step.
### Problem 62
\[
\left(\frac{15 a^{2} b^{-2}}{-3 a b^{-3}}\right)^{-2}
\]
First, simplify the fraction inside the parentheses:
\[
\frac{15 a^{2} b^{-2}}{-3 a b^{-3}} = \frac{15}{-3} \cdot \frac{a^{2}}{a} \cdot \frac{b^{-2}}{b^{-3}} = -5 \cdot a^{1} \cdot b^{1} = -5 a b
\]
Now, raise it to the power of -2:
\[
(-5 a b)^{-2} = \frac{1}{(-5 a b)^{2}} = \frac{1}{25 a^{2} b^{2}} = \frac{1}{25 a^{2} b^{2}}
\]
### Problem 63
\[
\left(x^{-3} y^{-1}\right)^{-1}\left(x^{-3} y^{0}\right)^{2}
\]
First, simplify each part:
\[
\left(x^{-3} y^{-1}\right)^{-1} = x^{3} y^{1} = x^{3} y
\]
\[
\left(x^{-3} y^{0}\right)^{2} = (x^{-3})^{2} (y^{0})^{2} = x^{-6} \cdot 1 = x^{-6}
\]
Now multiply the two results:
\[
x^{3} y \cdot x^{-6} = x^{3 - 6} y = x^{-3} y
\]
### Problem 65
\[
\left[\frac{\left(a^{3} b^{5}\right)^{2}}{a^{5} b^{2}}\right]^{-1}
\]
First, simplify the fraction:
\[
\frac{(a^{3} b^{5})^{2}}{a^{5} b^{2}} = \frac{a^{6} b^{10}}{a^{5} b^{2}} = a^{6-5} b^{10-2} = a^{1} b^{8}
\]
Now raise it to the power of -1:
\[
(a^{1} b^{8})^{-1} = a^{-1} b^{-8}
\]
### Problem 66
\[
\left(\frac{s^{-3}}{4 t}\right)^{-3}\left(\frac{5 t}{s^{-7}}\right)^{-2}
\]
First, simplify each part:
\[
\left(\frac{s^{-3}}{4 t}\right)^{-3} = \left(\frac{1}{4} s^{-3} t^{-1}\right)^{-3} = 4^{3} s^{9} t^{3} = 64 s^{9} t^{3}
\]
\[
\left(\frac{5 t}{s^{-7}}\right)^{-2} = \left(5 t s^{7}\right)^{-2} = \frac{1}{(5 t s^{7})^{2}} = \frac{1}{25 t^{2} s^{14}} = 25^{-1} t^{-2} s^{-14}
\]
Now multiply the two results:
\[
(64 s^{9} t^{3}) \cdot \left(\frac{1}{25 t^{2} s^{14}}\right) = \frac{64 s^{9} t^{3}}{25 t^{2} s^{14}} = \frac{64}{25} s^{9-14} t^{3-2} = \frac{64}{25} s^{-5} t^{1}
\]
### Problem 68
\[
\left[\left(\frac{x^{5} y^{2}}{x^{-3} y}\right)^{-2}\left(\frac{y^{-3}}{2 x^{5}}\right)^{3}\right]^{-1}
\]
First, simplify each part:
\[
\frac{x^{5} y^{2}}{x^{-3} y} = x^{5 - (-3)} y^{2 - 1} = x^{8} y^{1}
\]
Now raise it to the power of -2:
\[
(x^{8} y)^{-2} = x^{-16} y^{-2}
\]
Now simplify the second part:
\[
\left(\frac{y^{-3}}{2 x^{5}}\right)^{3} = \frac{(y^{-3})
Quick Answer
### Problem 62
\[
\frac{1}{25 a^{2} b^{2}}
\]
### Problem 63
\[
x^{-3} y
\]
### Problem 65
\[
a^{-1} b^{-8}
\]
### Problem 66
\[
\frac{64}{25} s^{-5} t
\]
### Problem 68
\[
\left(\frac{x^{5} y^{2}}{x^{-3} y}\right)^{-2} = x^{-16} y^{-2}
\]
\[
\left(\frac{y^{-3}}{2 x^{5}}\right)^{3} = \frac{y^{-9}}{8 x^{15}}
\]
\[
\left[\left(x^{-16} y^{-2}\right)\left(\frac{y^{-9}}{8 x^{15}}\right)\right]^{-1} = \left(\frac{x^{-16} y^{-11}}{8 x^{15}}\right)^{-1} = \left(\frac{y^{-11}}{8 x^{-1}}\right)^{-1} = \left(\frac{8 x}{y^{11}}\right)^{1} = \frac{8 x}{y^{11}}
\]
### Problem 69
\[
\left[\frac{\left(a^{-5} b^{2}\right)^{-1}}{\left(-a^{1} b^{4} c^{-1}\right)^{2}}\right]^{-3} = \left[\frac{a^{5} b^{-2}}{a^{2} b^{8} c^{2}}\right]^{-3} = \left(\frac{a^{3} b^{-10} c^{-2}}{1}\right)^{3} = a^{9} b^{-30} c^{-6}
\]
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