Question

Take a factor out of the square root:

a)
`\sqrt{x^(3) }`
b)
`\sqrt{a^(5) }`

Answer 1

(a)
`√(x^3)  = x√(x)`

(b)
`√(a^5)  = a^2√(a)`

Explanation:

Here, the given expression is :

`(a)  √(x^3)`

Now, cube of any number
`a = (a)^3  = a *  a  * a`

So,
`(x)^3  = x *  x  * x   = (x) ^2  * x`

`\implies   √(x^3)  =   √(x^2 * x)  = √(x^2)√(x)   = x√(x)`

(because
`√(a^2)    =  a`)

Hence,
`√(x^3)  = x√(x)`

Here, the given expression is :

`(b)  √(a^5)`

Now, five power of any number
`m = (m)^5  = m * m  * m * m  * m`

So,
`(x)^5  = x *  x  * x  * x * x   = (x) ^4  * x`

`\implies   √(a^5)  =   √(a^4 * x)  = √(a^4)√(a)   = a^2√(x)`

(because
`√(a^4)    =  a^2`)

Hence,
`√(a^5)  = a^2√(a)`