Simplify the exponential expression (x^5 y^6 z^3 / (x^-5 y^-6 z^-3))^-2. Assume the variables represent nonzero real numbers.

Step-by-step solution

1. 1

   Simplify inside the parentheses before applying the outer power. Two exponent rules are competing for attention here, and the order matters for keeping the signs straight. Collapse the fraction to a single monomial first, then deal with the $-2$ once. The tool for the fraction is the quotient rule:

   $\frac{a^{m}}{a^{n}}=a^{m-n}$

2. 2

   Subtract the exponents variable by variable, watching the double negatives. Each denominator exponent is negative, so subtracting it adds:

   $\frac{x^{5}}{x^{-5}}=x^{5-(-5)}=x^{10},\qquad \frac{y^{6}}{y^{-6}}=y^{6-(-6)}=y^{12},\qquad \frac{z^{3}}{z^{-3}}=z^{3-(-3)}=z^{6}$

   So the expression is now

   $\left(x^{10}y^{12}z^{6}\right)^{-2}$

3. 3

   Distribute the outer exponent with the power rule. Since $\left(a^{m}\right)^{n}=a^{mn}$, and a product raised to a power raises each factor, every exponent gets multiplied by $-2$:

   $x^{10(-2)}y^{12(-2)}z^{6(-2)}=x^{-20}y^{-24}z^{-12}$

4. 4

   Convert the negative exponents into a positive-exponent form. A negative exponent means a reciprocal, $a^{-k}=\dfrac{1}{a^{k}}$, so all three factors drop into one denominator:

   $x^{-20}y^{-24}z^{-12}=\frac{1}{x^{20}y^{24}z^{12}}$

   This is why the problem insists the variables are nonzero — otherwise the reciprocal would be undefined.

5. 5

   Verify numerically. Substituting $x=1.3$, $y=0.7$, $z=2.1$ into the original expression gives $0.0037337776928081$, and the simplified form $1/(x^{20}y^{24}z^{12})$ gives the same $0.0037337776928081$. The two agree to full double precision, so the exponent bookkeeping is correct.