Solving Exponential Equations: Finding the Product $xyz$

We are given three equations involving exponents and variables $p, q, r, x, y, z$. We need to find the value of the product $xyz$. The condition $pqr \neq 0$ ensures that the bases and their reciprocals are well-defined and non-zero.

Step 1: Rewrite the given equations

Let's simplify the given equations using the properties of exponents ($a^{-n} = 1/a^n$):

• $p^{-x} = \frac{1}{q}$ implies $\frac{1}{p^x} = \frac{1}{q}$. Taking the reciprocal of both sides gives $p^x = q$.
• $q^{-y} = \frac{1}{r}$ implies $\frac{1}{q^y} = \frac{1}{r}$. Taking the reciprocal of both sides gives $q^y = r$.
• $r^{-z} = \frac{1}{p}$ implies $\frac{1}{r^z} = \frac{1}{p}$. Taking the reciprocal of both sides gives $r^z = p$.

Step 2: Substitute equations to find a relationship

We can use substitution to relate $p, x, y, z$. Start with the first simplified equation $p^x = q$. Substitute this into the second simplified equation $q^y = r$:

$(p^x)^y = r$

Using the power of a power rule ($(a^m)^n = a^{mn}$), we get:

$p^{xy} = r$

Now, substitute this result ($r = p^{xy}$) into the third simplified equation $r^z = p$:

$(p^{xy})^z = p$

Again, using the power of a power rule:

$p^{xyz} = p$

Step 3: Determine the value of $xyz$

The equation $p^{xyz} = p$ can be written as $p^{xyz} = p^1$. Since we are given that $pqr \neq 0$, we know $p \neq 0$. If $p=1$, the original equations become trivial ($1=1$). Assuming $p \neq 1$ (and $p \neq -1$ for generality, though the logic holds), we can equate the exponents:

$xyz = 1$

Therefore, the value of the product $xyz$ is 1.