Pregunta
Simplify the expression \( \left(\frac{1}{4 a b}\right)^{-2} \). Assume \( a \neq 0, b \neq 0 \)
Ask by Olson Gough. in the United States
Mar 10,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
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The simplified expression is \( 16a^2b^2 \).
Solución
1. Start with the expression:
\[
\left(\frac{1}{4ab}\right)^{-2}
\]
2. Recall that for any nonzero \( x \) and integer \( n \),
\[
x^{-n} = \frac{1}{x^n}
\]
However, applying the rule in reverse, we have:
\[
\left(\frac{1}{x}\right)^{-n} = x^n
\]
In this case, \( x = 4ab \) and \( n = 2 \), so:
\[
\left(\frac{1}{4ab}\right)^{-2} = (4ab)^2
\]
3. Now, square the product:
\[
(4ab)^2 = 4^2 \cdot a^2 \cdot b^2 = 16a^2b^2
\]
The simplified expression is:
\[
16a^2b^2
\]
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Simplifique esta solución Extra Insights
To simplify the expression \( \left(\frac{1}{4 a b}\right)^{-2} \), we can apply the property of exponents that states \( x^{-n} = \frac{1}{x^n} \). Starting with our expression: \[ \left(\frac{1}{4 a b}\right)^{-2} = \frac{1}{\left(\frac{1}{4 a b}\right)^{2}} \] Now, rewriting \( \left(\frac{1}{4 a b}\right)^{2} \): \[ \left(\frac{1}{4 a b}\right)^{2} = \frac{1^2}{(4 a b)^{2}} = \frac{1}{16 a^2 b^2} \] Thus, substituting this back, we get: \[ \frac{1}{\left(\frac{1}{4 a b}\right)^{2}} = \frac{1}{\frac{1}{16 a^2 b^2}} = 16 a^2 b^2 \] Therefore, the simplified expression is: \[ \boxed{16 a^2 b^2} \]
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