Question

How do you simplify `((4a^2b)/(a^3b^2))((5a^2b)/(2b^4))` and write it using only positive exponents?

Answer 1

See a solution process below:

Explanation:

First, rewrite this expression as:

`(4 * 5)/2((a^2 * a^2)/a^3)((b * b)/(b^2 * b^4)) =>`

`20/2((a^2 * a^2)/a^3)((b * b)/(b^2 * b^4)) =>`

`10((a^2 * a^2)/a^3)((b * b)/(b^2 * b^4))`

Next, use this rule of exponents to rewrite the numerator for the `b` terms:

`a = a^color(red)(1)`

`10((a^2 * a^2)/a^3)((b^color(red)(1) * b^color(red)(1))/(b^2 * b^4))`

Then, use this rule of exponents to multiply the numerators for both the `a` and `b` terms and the denominator for the `b` term:

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

`10((a^color(red)(2) * a^color(blue)(2))/a^3)((b^color(red)(1) * b^color(blue)(1))/(b^color(red)(2) * b^color(blue)(4))) =>`

`10(a^(color(red)(2)+color(blue)(2))/a^3)(b^(color(red)(1)+color(blue)(1))/b^(color(red)(2)+color(blue)(4))) =>`

`10(a^4/a^3)(b^2/b^6)`

Next, use these rules of exponents to complete the simplification of the `a` term:

`x^color(red)(a)/x^color(blue)(b) = x^(color(red)(a)-color(blue)(b))` and `a^color(red)(1) = a`

`10(a^color(red)(4)/a^color(blue)(3))(b^2/b^6) =>`

`10a^(color(red)(4)-color(blue)(3))(b^2/b^6) =>`

`10a^color(red)(1)(b^2/b^6) =>`

`10a(b^2/b^6)`

Now, use this rule of exponents to complete the simplification for the `b` term:

`x^color(red)(a)/x^color(blue)(b) = 1/x^(color(blue)(b)-color(red)(a))`

`10a(b^color(red)(2)/b^color(blue)(6)) =>`

`10a(1/b^(color(blue)(6)-color(red)(2))) =>`

`10a(1/b^4) =>`

`(10a)/b^4`