Simplify (a^2b^8c^5)^2(a^5c^2)^3(a^-2bc^-1)^3

Problem

(a2*b8*c5)2*(a5*c2)3*(a(−2)*b*c(−1))3

Solution

1. Apply the power of a power rule (xm)n=x(m*n) to each set of parentheses by multiplying the outer exponent by every exponent inside.

(a(2⋅2)*b(8⋅2)*c(5⋅2))*(a(5⋅3)*c(2⋅3))*(a(−2⋅3)*b(1⋅3)*c(−1⋅3))

2. Calculate the products of the exponents for each variable.

(a4*b16*c10)*(a15*c6)*(a(−6)*b3*c(−3))

3. Group the terms with the same base together.

(a4*a15*a(−6))*(b16*b3)*(c10*c6*c(−3))

4. Apply the product rule xm*xn=x(m+n) by adding the exponents for each base.

a(4+15−6)*b(16+3)*c(10+6−3)

5. Simplify the sums in the exponents to find the final powers.

a13*b19*c13

Final Answer

(a2*b8*c5)2*(a5*c2)3*(a(−2)*b*c(−1))3=a13*b19*c13

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