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}$.

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Updated on: 10-Oct-2022

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