Question

Let x,y,z be positive real numbers , simplify each of the following - √x^-2y^3​

Answer 1

Given:

$\sqrt{x^{-2}y^3}$

where, x,y,z are positive real numbers.

To find:

The simplified form of the given expression.

Solution:

We have,

$\sqrt{x^{-2}y^3}$

Using the properties of exponents and radical, we get

$=\sqrt{x^{-2}}\cdot \sqrt{y^3}$    $[\because \sqrt{ab}=\sqrt{a}\sqrt{b}]$

$=\sqrt{\dfrac{1}{x^{2}}}\cdot \sqrt{y^{2+1}}$    $[\because x^{-n}=\dfrac{1}{x^n}]$

$=\sqrt{\left(\dfrac{1}{x}\right)^2}\cdot \sqrt{y^2\cdot y}$    $[\because a^{m+n}=a^ma^n]$

$=\sqrt{\left(\dfrac{1}{x}\right)^2}\cdot \sqrt{y^2}\cdot \sqrt{y}$    $[\because \sqrt{ab}=\sqrt{a}\sqrt{b}]$

$=\dfrac{1}{x}\cdot y\sqrt{y}$

$=\dfrac{y\sqrt{y}}{x}$

Therefore, the simplified form of given expression is $\dfrac{y\sqrt{y}}{x}$.