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Simplify \( \left[\left\{\left(\frac{1}{2}\right)^{2}\right\}^{-2}\right]^{-2} \).
Given:
$ \begin{array}{l} \left[\left\{\left(\frac{1}{2}\right)^{2}\right\}^{-2}\right]^{-2}\ \ \end{array}$
To do:
We have to simplify the given expression.
Solution:
We know that,
$\left( a^{m}\right)^{n} =a^{m\times n}$
$a^{-m} =\frac{1}{a^{m}}$
Therefore,
$\left[\left\{\left(\frac{1}{2}\right)^{2}\right\}^{-2}\right]^{-2}=\left[\left(\frac{1}{2}\right)^{2\times ( -2)}\right]^{-2}$
$=\left[\left(\frac{1}{2}\right)^{-4}\right]^{-2}$ [( +) \times ( -) =( -)]
$=\left(\frac{1}{2}\right)^{( -4) \times ( -2)}$
$=\left(\frac{1}{2}\right)^{8}$ [( -) \times ( -) =( +)]
$=\frac{1}{256}$.
$ \begin{array}{l} \left[\left\{\left(\frac{1}{2}\right)^{2}\right\}^{-2}\right]^{-2}\ \ \end{array}$$=\frac{1}{256}$.