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

Answer:

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#