Question

Simplify the following? `(x^(5/4) sqrt(16x^3)) /(8x)^(2/3)`

Answer 1

`x^(25/12)`

Explanation:

I think the question here is to simplify the following:

`(x^(5/4)sqrt(16x^3))/(8x)^(2/3)`

I'll first move the term out of the denominator. Remember that `1/x=x^(-1)`, so we can rewrite the denominator:

`1/(8x)^(2/3)=(8x)^(-2/3)`

and we can simplify further:

`(8x)^(-2/3)=8^(-2/3)x^(-2/3)=(2^3)^(-2/3)x^(-2/3)=2^(3xx(-2/3))x^(-2/3)=2^-2x^(-2/3)`

We can now write the original as:

`(x^(5/4)sqrt(16x^3))/(8x)^(2/3)=(x^(5/4))(sqrt(16x^3))(2^-2x^(-2/3))`

Let's rewrite the middle term:

`sqrt(16x^3)=16^(1/2)x^(3/2)=4x^(3/2)`

And substitue in:

`(x^(5/4))(4x^(3/2))(2^-2x^(-2/3))`

Let's now do the multiplication. I'll group the constants together and the variables together:

`(4)(2^-2)xx(x^(5/4))(x^(3/2))(x^(-2/3))`

For the constants, we have:

`4xx2^-2=4xx1/4=1`

For the variables, remember that we can use the rule

`x^a xx x^b = x^(a+b)`

We have:

`(x^(5/4))(x^(3/2))(x^(-2/3))=x^(5/4+3/2-2/3)=x^(15/12+18/12-8/12)=x^(25/12)`