Question

How do you simplify `[(4x^-4 y^2)/ (5x^6 y^-3)]^-3` leaving only positive exponents?

Answer 1

`(125x^30)/(64y^15)`

Explanation:

Given,

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

According to the exponent law , `(xy)^n=x^ny^n`, the `color(blue)(-3)` multiplies by each exponent in the numerator and denominator. In addition, the numbers are raised to `color(blue)(-3)`. Thus,

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

Simplifying,

`=(4^-3x^12y^-6)/(5^-3x^-18y^9)`

According to the exponent law , `x^-n/1=1/x^n`. Similarly, `1/x^-n=x^n/1`. Thus,

`=(5^3x^(12)*5x^18)/(4^3y^9y^6)`

Simplifying,

`=(125x^(12)x^18)/(64y^9y^6)`

Using the exponent law , `x^mx^n=x^(m+n)`, the exponents in the numerator of base `x` can be added together. Similarly, the exponents in the denominator of base `y` can be added together.

`=(125x^(12+18))/(64y^(9+6))`

`=color(green)(|bar(ul(color(white)(a/a)color(black)((125x^30)/(64y^15))color(white)(a/a)|)))`