Simplify ((xy^-2z^-3)/(x^2y^3z^-4))^-3

Problem

((x*y^(−2)*z^(−3))/(x^2*y^3*z^(−4)))^(−3)

Solution

1. Simplify the expression inside the parentheses by applying the quotient rule for exponents, a^m/a^n=a^(m−n) for each variable.

(x^1)/(x^2)=x^(1−2)=x^(−1)

(y^(−2))/(y^3)=y^(−2−3)=y^(−5)

(z^(−3))/(z^(−4))=z^(−3−(−4))=z^1

2. Rewrite the inner expression using the simplified variables.

(x^(−1)*y^(−5)*z^1)^(−3)

3. Apply the power of a power rule, (a^m)^n=a^(m⋅n) by multiplying each exponent inside the parentheses by −3

x^(−1⋅−3)=x^3

y^(−5⋅−3)=y^15

z^(1⋅−3)=z^(−3)

4. Express with positive exponents by moving the variable with the negative exponent to the denominator.

x^3*y^15*z^(−3)=(x^3*y^15)/(z^3)

Final Answer

((x*y^(−2)*z^(−3))/(x^2*y^3*z^(−4)))^(−3)=(x^3*y^15)/(z^3)

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