Question
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Tutor-Verified Answer
Answer
\( 16x^{4}y^{-4} \)
Solution
1. Expandir la expresión:
\[
\left(4 x^{2} y^{-2}\right)^{2} = 4^{2} \cdot \left(x^{2}\right)^{2} \cdot \left(y^{-2}\right)^{2}
\]
2. Aplicar la potencia a cada factor:
\[
4^{2} = 16, \quad \left(x^{2}\right)^{2} = x^{2 \cdot 2} = x^{4}, \quad \left(y^{-2}\right)^{2} = y^{-2 \cdot 2} = y^{-4}
\]
3. Escribir el resultado final:
\[
16 x^{4} y^{-4}
\]
Si se prefiere escribir el resultado sin exponentes negativos, se puede expresar como:
\[
16 \frac{x^{4}}{y^{4}}
\]
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Simplify this solution Beyond the Answer
To simplify the expression \( \left(4 x^{2} y^{-2}\right)^{2} \), let's apply the power of a product rule. Each factor inside the parentheses is raised to the power of 2. That means we'll raise 4, \( x^{2} \), and \( y^{-2} \) individually: 1. \( 4^{2} = 16 \) 2. \( (x^{2})^{2} = x^{4} \) (by multiplying the exponents) 3. \( (y^{-2})^{2} = y^{-4} \) Now, putting it all together: \( (4 x^{2} y^{-2})^{2} = 16 x^{4} y^{-4} \) This can be expressed with positive exponents in the denominator, giving: \( 16 x^{4} \cdot \frac{1}{y^{4}} = \frac{16 x^{4}}{y^{4}} \) So the final simplified result is: \(\frac{16 x^{4}}{y^{4}}\)
