Question
Which is the simplified form of the expression (StartFraction (2 Superscript negative 3 Baseline) (x Superscript negative 3 Baseline) (y squared) Over (4 Superscript negative 2 Baseline) (x Superscript 4 Baseline) (y Superscript 6 Baseline) EndFraction) Squared?
StartFraction 1 Over 4 x squared y Superscript 16 Baseline EndFraction
StartFraction 4 Over x squared y Superscript 16 Baseline EndFraction
StartFraction 1 Over 4 x Superscript 14 y Superscript 8 Baseline EndFraction
StartFraction 4 Over x Superscript 14 Baseline y Superscript 8 Baseline EndFraction
StartFraction 1 Over 4 x squared y Superscript 16 Baseline EndFraction
StartFraction 4 Over x squared y Superscript 16 Baseline EndFraction
StartFraction 1 Over 4 x Superscript 14 y Superscript 8 Baseline EndFraction
StartFraction 4 Over x Superscript 14 Baseline y Superscript 8 Baseline EndFraction
Answers
Answered by
GPT-4o mini
To simplify the expression
\[
\left( \frac{(2^{-3})(x^{-3})(y^2)}{(4^{-2})(x^4)(y^6)} \right)^2,
\]
let's first simplify the fraction inside the parentheses before squaring it.
1. **Rewrite the bases**:
Since \(4^{-2} = (2^2)^{-2} = 2^{-4}\), we have:
\[
4^{-2} = 2^{-4}.
\]
Thus, we can rewrite the expression as:
\[
\frac{(2^{-3})(x^{-3})(y^2)}{(2^{-4})(x^4)(y^6)}.
\]
2. **Simplify by separating terms**:
We can now simplify the expression term by term:
\[
\frac{2^{-3}}{2^{-4}} = 2^{-3 - (-4)} = 2^{-3 + 4} = 2^{1} = 2,
\]
\[
\frac{x^{-3}}{x^{4}} = x^{-3 - 4} = x^{-7},
\]
\[
\frac{y^{2}}{y^{6}} = y^{2 - 6} = y^{-4}.
\]
Thus, we have:
\[
\frac{(2)(x^{-7})(y^{-4})} = 2x^{-7}y^{-4}.
\]
3. **Combine into a single fraction**:
We can express this as:
\[
\frac{2}{x^{7}y^{4}}.
\]
4. **Square the whole expression**:
Now we take the squared result:
\[
\left(\frac{2}{x^{7}y^{4}}\right)^2 = \frac{2^2}{(x^{7})^2(y^{4})^2} = \frac{4}{x^{14}y^{8}}.
\]
Thus, the simplified form of the expression is:
\[
\frac{4}{x^{14}y^{8}}.
\]
Therefore, the correct answer is
\[
\boxed{\frac{4}{x^{14}y^{8}}}.
\]
\[
\left( \frac{(2^{-3})(x^{-3})(y^2)}{(4^{-2})(x^4)(y^6)} \right)^2,
\]
let's first simplify the fraction inside the parentheses before squaring it.
1. **Rewrite the bases**:
Since \(4^{-2} = (2^2)^{-2} = 2^{-4}\), we have:
\[
4^{-2} = 2^{-4}.
\]
Thus, we can rewrite the expression as:
\[
\frac{(2^{-3})(x^{-3})(y^2)}{(2^{-4})(x^4)(y^6)}.
\]
2. **Simplify by separating terms**:
We can now simplify the expression term by term:
\[
\frac{2^{-3}}{2^{-4}} = 2^{-3 - (-4)} = 2^{-3 + 4} = 2^{1} = 2,
\]
\[
\frac{x^{-3}}{x^{4}} = x^{-3 - 4} = x^{-7},
\]
\[
\frac{y^{2}}{y^{6}} = y^{2 - 6} = y^{-4}.
\]
Thus, we have:
\[
\frac{(2)(x^{-7})(y^{-4})} = 2x^{-7}y^{-4}.
\]
3. **Combine into a single fraction**:
We can express this as:
\[
\frac{2}{x^{7}y^{4}}.
\]
4. **Square the whole expression**:
Now we take the squared result:
\[
\left(\frac{2}{x^{7}y^{4}}\right)^2 = \frac{2^2}{(x^{7})^2(y^{4})^2} = \frac{4}{x^{14}y^{8}}.
\]
Thus, the simplified form of the expression is:
\[
\frac{4}{x^{14}y^{8}}.
\]
Therefore, the correct answer is
\[
\boxed{\frac{4}{x^{14}y^{8}}}.
\]
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