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

Question

Genesis Allen · Accepted Answer

See a solution process below:

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#

Avery Alexander · Answer

undefined To simplify the expression ((4a^2b)/(a^3b^2))((5a^2b)/(2b^4)), first, multiply the numerators together and the denominators together separately. This results in (4a^2b * 5a^2b) / (a^3b^2 * 2b^4). Then, simplify each term in the numerator and denominator. After simplification, the expression becomes (20a^4b^2) / (2a^3b^6). Finally, divide the coefficients and subtract the exponents. The simplified expression is 10a^(4-3)b^(2-6), which equals 10a^(1)b^(-4).