Answer:
First "order of operations" mistake: step 2
First arithmetic mistake: step 4
Step-by-step explanation:
As we understand Rena's work, she wants to simplify ...
[tex]\left(\dfrac{x^{-3}y^{-2}}{2x^4y^{-4}}\right)^{-3}[/tex]
for x = -1 and y = 2.
Her work seems to be ...
Step 1
[tex]\text{Substitute $x=-1$ and $y=2$ into the expression}\\\\\left(\dfrac{(-1)^{-3}2^{-2}}{2(-1)^42^{-4}}\right)^{-3}\qquad\text{no error}[/tex]
Step 2
[tex]\text{Simplify the parentheses}\\\\\left(\dfrac{2^4}{2(-1)^4(-1)^32^2}\right)^{-3}=\left(\dfrac{2^2}{2(-1)^7}\right)^{-3}\qquad\text{order of operations error}[/tex]
Step 3
[tex]\text{Evaluate the power to a power}\\\\\dfrac{2^{-6}}{2^{-3}(-1)^{21}}\qquad\text{no error}[/tex]
Step 4
[tex]\text{Use reciprocals and find the value}\\\\\dfrac{1}{2^32^6(-1)^{21}}=\dfrac{1}{8\cdot 64\cdot (-1)}=\dfrac{-1}{512}\qquad\text{error: $2^3$ is used instead of $2^{-3}$}[/tex]
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So, the first arithmetic error is in Step 4. However, the order of operations requires exponents be evaluated first. Doing that makes step 2 look like ...
[tex]\left(\dfrac{-\dfrac{1}{4}}{2(1)\dfrac{1}{16}}\right)^{-3}=(-2)^{-3}\qquad\text{proper Step 2}[/tex]
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We expect your answer is supposed to be Step 4.
