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,

$\sqrt{(\frac{x+1}{2})^{2}-(\frac{x-1}{2})^{2}}=\sqrt{(\frac{(x+1)^2}{2^2}-\frac{(x-1)^2}{2^2}}$

$=\sqrt{\frac{x^2+2\times x \times1+1^2}{4}-\frac{x^2-2\times x \times1+1^2}{4}}$

$=\sqrt{\frac{x^2+2x+1-x^2+2x-1}{4}}$

$=\sqrt{\frac{4x}{4}}$

$=\sqrt{x}$