How do you multiply #(25a^2b)^3(1/5abc)^2#?

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Answer 1 · 2015-04-22T03:49:10

Start with the exponent rule #(x^m)^n=x^(m*n)# .

#(25a^2b)^3(1/5abc)^2#=

#(25^3a^6b^3)((1/5)^2a^2b^2c^2)#=

#(15625a^6b^3)((1/25)a^2b^2c^2)#=

Divide 15625 by 25 and use the exponent rule #(x^m)(x^n)=x^(m+n)#.

#(15625/25)(a^(6+2))(b^(3+2))(c^2)=625a^8b^5c^2#

Answer 2 · 2015-04-22T03:58:29

We know that #color(blue)((ab)^2 = a^2 * b ^2#

Hence
#(25a^2b)^3(1/5abc)^2#

# = {25^3 * (a^2)^3 * b^3} * {(1/5)^2 * a^2 * b^2 * c^2}#

# = {(5^2)^3 * a^6 * b^3} * {(1/5)^2 * a^2 * b^2 * c^2}#

# = {5^6 * a^6 * b^3} * {(1/5)^2 * a^2 * b^2 * c^2}#

Next, we group the Constants and the Same Variables together

# = (5^6*(1/5)^2)*(a^6*a^2)*(b^3*b^2)*c^2#

# = (5^6/5^2)*(a^6*a^2)*(b^3*b^2)*c^2#

Two important laws of Exponents:

#color(green)(a^m*a^n = a^(m+n) if a!=0#
#color(green)(a^m/a^n = a^(m -n) if a!=0#

Applying these, we get

# = (5^(6-2))(a^(6+2))*(b^(3+2))*c^2#

# = 5^4 * a^8 * b^5 *c^2#

# = 625a^8b^5c^2#