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

1 Answer
Jul 31, 2017

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