Simplify (a^3b^12c^2)^2(a^5c^2)^3(b^5c^4)^0

Problem

(a3*b12*c2)2*(a5*c2)3*(b5*c4)0

Solution

1. Apply the zero exponent rule to the third term, noting that any non-zero expression raised to the power of 0 equals 1

(b5*c4)0=1

2. Apply the power of a power rule (xm)n=x(m*n) to the first and second terms by multiplying the outer exponents with the inner exponents.

(a3*b12*c2)2=a(3⋅2)*b(12⋅2)*c(2⋅2)=a6*b24*c4

(a5*c2)3=a(5⋅3)*c(2⋅3)=a15*c6

3. Combine the expressions by multiplying the results from the previous steps.

(a6*b24*c4)*(a15*c6)*(1)

4. Apply the product rule xm⋅xn=x(m+n) to combine terms with the same base by adding their exponents.

a(6+15)*b24*c(4+6)

5. Simplify the exponents to find the final expression.

a21*b24*c10

Final Answer

(a3*b12*c2)2*(a5*c2)3*(b5*c4)0=a21*b24*c10

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