Question

How do you multiply and simplify `\frac { 5x ^ { 4} } { 10y ^ { 2} } \cdot \frac { y ^ { 7} x } { x y }`?

Answer 1

First, rewrite this expression as:

`(5/10)((x^4 * x)/x)(y^7/(y^2 * y)) ->`

`(1/2)((x^4 * color(red)(cancel(color(black)(x))))/color(red)(cancel(color(black)(x))))(y^7/(y^2 * y)) ->`

`1/2x^4(y^7/(y^2 * y))`

Next, we can use these rules of exponents to simplify the denominator of the `y` terms:

`a = a^color(blue)(1)` and `x^color(red)(a) xx x^color(blue)(b) = x^(color(red)(a) + color(blue)(b))`

`1/2x^4(y^7/(y^color(red)(2) * y^color(blue)(1))) ->`

`1/2x^4(y^7/(y^(color(red)(2)+color(blue)(1)))) ->`

`1/2x^4(y^7/(y^3))`

Now, we can use this rule of exponents to complete the simplification of the `y` terms:

`x^color(red)(a)/x^color(blue)(b) = x^(color(red)(a)-color(blue)(b))`

`1/2x^4(y^color(red)(7)/y^color(blue)(3)) ->`

`1/2x^4y^(color(red)(7)-color(blue)(3)) ->`

`1/2x^4y^4`

Or

`(x^4y^4)/2`