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Simplify:$ \sqrt{\left(\frac{x+1}{2}\right)^{2}-\left(\frac{x-1}{2}\right)^{2}} $
Given:
\( \sqrt{\left(\frac{x+1}{2}\right)^{2}-\left(\frac{x-1}{2}\right)^{2}} \)
To do:
We have to simplify \( \sqrt{\left(\frac{x+1}{2}\right)^{2}-\left(\frac{x-1}{2}\right)^{2}} \).
Solution:
We know that,
$(a^{m})^{n}=a^{m n}$
$a^{m} \times a^{n}=a^{m+n}$
$a^{m} \div a^{n}=a^{m-n}$
$a^{0}=1$
Therefore,
$\sqrt{(\frac{x+1}{2})^{2}-(\frac{x-1}{2})^{2}}=\sqrt{(\frac{(x+1)^2}{2^2}-\frac{(x-1)^2}{2^2}}$
$=\sqrt{\frac{x^2+2\times x \times1+1^2}{4}-\frac{x^2-2\times x \times1+1^2}{4}}$
$=\sqrt{\frac{x^2+2x+1-x^2+2x-1}{4}}$
$=\sqrt{\frac{4x}{4}}$
$=\sqrt{x}$
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