To evaluate whether Ben simplified the expression correctly, let’s start with the expression he began with:
\[ \left( \frac{2x^{-4}y^7}{3x^3y^2} \right)^3 \]
We can simplify the expression inside the parentheses first.
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Simplifying the fractions:
The numerator is \(2x^{-4}y^7\) and the denominator is \(3x^3y^2\).
We can separate the components:
- For constants: \(\frac{2}{3}\)
- For \(x\): \(\frac{x^{-4}}{x^3} = x^{-4 - 3} = x^{-7}\) (add the exponents)
- For \(y\): \(\frac{y^7}{y^2} = y^{7 - 2} = y^{5}\) (subtract the exponents)
So, the expression simplifies to:
\[ \frac{2}{3} x^{-7} y^5 \]
Now, substituting this back into the original expression gives us:
\[ \left( \frac{2}{3} x^{-7} y^5 \right)^3 \]
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Applying the exponent:
Now raise each part of the fraction to the third power:
- For the constant: \(\left(\frac{2}{3}\right)^3 = \frac{8}{27}\)
- For \(x^{-7}\): \((x^{-7})^3 = x^{-21}\) (multiply the exponents)
- For \(y^{5}\): \((y^{5})^3 = y^{15}\) (multiply the exponents)
Now combining these together gives:
\[ \frac{8y^{15}}{27x^{21}} \]
Based on the options provided:
- \(\frac{2y^5}{3x^7}\) cubed is incorrect since this doesn’t match our calculated values.
- \(\frac{2y^{15}}{3x^{21}}\) is not correct either as our calculated value is \(\frac{8y^{15}}{27x^{21}}\).
Now, going back to your original question regarding Ben's work:
- First evaluation: Yes, he needed to add the exponents of powers of the same base in the first step.
- Second evaluation: No, he needed to apply the exponent to all factors in the product in the second step.
If Ben made a mistake, it would be related to not correctly applying the exponent to all factors after simplification and correctly handling the exponents in the initial simplification.
The correct answer is No, he needed to apply the exponent to all factors in the product in the second step.