How do you simplify #(3ab^2c)^2(-2a^2b^4)^2(a^4c^2)^3(a^2b^4c^5)^2(2a^3b^2c^4)^3#?

We are given

#[(3ab^2c)^2][(-2a^2b^4)^2][(a^4c^2)^3][(a^2b^4c^5)^2][(2a^3b^2c^4)^3]#

We will simplify each bracket separately for better understanding.

We will also rewrite each factor by associating exponents to every element of the terms, before simplification.

#(3ab^2c)^2# bracket 1

#= (3)^2 (a)^2 (b^2)^2 (c)^2#

#= 9a^2b^4c^2# {Exponent rule used: #(a ^ m)^n = a^(mn)]#

#(-2a^2b^4)^2# bracket 2

#=(-2)^2(a^2)^2(b^4)^2#

#= 4 a^4 b^8# {Exponent rule used: #(a ^ m)^n = a^(mn)]#

#(a^4c^2)^3# bracket 3

#= (a^4)^3(c^2)^3#

#=a^12c^6# {Exponent rule used: #(a ^ m)^n = a^(mn)]#

#(a^2b^4c^5)^2# bracket 4

#=(a^2)^2(b^4)^2(c^5)^2#

#=a^4 b ^ 8 c^10# {Exponent rule used: #(a ^ m)^n = a^(mn)]#

#(2a^3b^2c^4)^3# bracket 5

#=(2^3)(a^3)^3(b^2)^3 (c^4)^3#

#8 a^9 b ^ 6 c^ 12# {Exponent rule used: #(a ^ m)^n = a^(mn)]#

Our next step is to write the simplified factors as one product for further simplification.

#(9a^2b^4c^2)(4 a^4 b^8)(a^12c^6)(a^4 b ^ 8 c^10)(8 a^9 b ^ 6 c^ 12)#

#=288a^31b^26c^30# {Exponent rule used: #(a ^ m)( a ^ n) = a^(m+n)]#

This is our final answer after simplification.

#(3ab^2c)^2(−2a^2b^4)^2(a^4c^2)^3(a^2b^4c^5)^2 (2a^3b^2c^4)^3#

Raise every exponent inside a parentheses to the power outside.

To raise a power to a power, you multiply the exponents.

Don't forget to include exponents which are an unwritten 1.

After you multiply all the exponents inside a parentheses by the exponent outside it, you will have this:

#(3^2  a^2  b^4  c^2)(-2^2  a^4  b^8)(a^12  c^6)(a^4  b^8  c^10)(2^3  a^9  b^6  c^12)#
....................................

Now multiply all the like bases together -- all the a's get multiplied to each other, then all the b's, and then all the c's.
The numbers also are multiplied together.

This is much easier if you group the like bases together first.

#(3^2)(-2^2)(2^3) xx (a^2* a^4*a^12*a^4*a^9)xx (b^4* b^8*b^8*b^6)xx (c^2*c^6*c^10* c^12)#

Now you can multiply all the a's, all the b's, etc by adding their exponents.

To multiply exponents with like bases, you add the exponents.
#(9*4*8)(a  ^31)(b  ^26)(c ^30)#

Answer:
#288   a ^31  b ^26  c ^30#