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 √(y^3)`
`[\because √(ab)=√(a)√(b)]`

`=\sqrt{(1)/(x^(2))}\cdot \sqrt{y^(2+1)}`
`[\because x^(-n)=(1)/(x^n)]`

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

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

`=(1)/(x)\cdot y√(y)`

`=(y√(y))/(x)`

Therefore, the simplified form of given expression is
`(y√(y))/(x)`.