Question

How do you simplify `[(8(x^3) ^2 y^-3)/(6y^2 x^-2)] [(4(x^2 ) ^2 y^-3)/(12x^-3 y^2 )]` using only positive exponents?

Answer 1

`(4x^15)/(9y^10)`

Explanation:

Using the following `color(blue)" rules of exponents "`

`• a^mxxa^n = a^(m+n)`

example : `2^2xx2^3 = 2^(2+3) = 2^5 = 32`
`color(red)"-------------------------------------------------"`
`•( a^m)/(a^n) = a^(m-n)`
example`(3^4)/(3^2) = 3^(4-2)=3^2=9`
and `(2^3)/(2^5) = 2^(3-5)=2^(-2)`
`color(red)"--------------------------------------------------"`
`• a^(-m) hArr 1/(a^m)`
example (from above): `(2^3)/(2^5)=2^(-2)=1/(2^2)=1/4`
`color(red)"----------------------------------------------------"`
`• (a^m)^n = a^(mxxn)= a^(mn)`

example`(2^2)^3=2^(2xx3)=2^6=64`
`color(red)"-----------------------------------------------------"`

now applying these to the question.

first bracket`[(8(x^3)^2y^(-3))/(6y^2x^-2)]=8/6xx(x^6)/(x^-2)xx(y^-3)/y^2`

`=4/3x^(6-(-2))y^(-3-2)=4/3 x^8 y^-5`
`color(blue)"-------------------------------------------------"`

second bracket `[(4(x^2)^2y^-3)/(12x^-3y^2)]=4/12xxx^4/x^-3xxy^-3/y^2`

`=1/3x^7y^-5`
`color(blue)"------------------------------------------------"`

Now multiplying the 2 simplifications together

`4/3 x^8y^-5xx1/3 x^7y^-5 =4/9x^(15)y^(-10)=(4x^(15))/(9y^(10)`