How do you simplify #(9p^2(3pg)^3)/(81g^(-2)) #?

1 Answer
Lil Del
Oct 9, 2016

#(3p^5g^5)#

Explanation:

Several of the Properties of Exponents must be applied in order to simplify this expression. The Power of a Product Property states that if a product is raised to a power, each factor is individually raised to the power.
#(ab)^m = a^mb^m#
So, #(3pg)^3 = 3^3p^3g^3 = 27p^3g^3#.

The Product of Powers Property states that if like bases are being multiplied, the base remains the same and the exponents are added.
a^m*a^n = a^(m+n)# #9p^2 * 27p^3g^3 = 243p^5g^3#

The Quotient of Powers Property states that if like bases are being divided, the base remains the same and the powers are subtracted.
#a^m/a^n = a^(m - n)#
So, #g^3/g^-2 = g^(3 - -2) = g^5#

The numerical potion of the fraction, #243/81#, is simplified by division. #243 -: 81 = 3#.

This completes the simplification and yields the fully simplified answer: #3p^5g^5#

#9p^2 (3pg)^3/(81g^-2)#
#9p^2 (27p^3g^3)/(81g^-2)#
#(243p^5g^3)/(81g^-2)#
#(243p^5g^5)/81#
#3p^5g^5#

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