Sure! Let's solve the given equation step by step:
[tex]\[
\frac{x + y + z}{x^{-1} y^{-1} + y^{-1} z^{-1} + z^{-1} x^{-1}} = x y z
\][/tex]
### Step 1: Simplify the Denominator
First, we need to simplify the denominator [tex]\( x^{-1} y^{-1} + y^{-1} z^{-1} + z^{-1} x^{-1} \)[/tex]:
[tex]\[
x^{-1} y^{-1} = \frac{1}{xy}
\][/tex]
[tex]\[
y^{-1} z^{-1} = \frac{1}{yz}
\][/tex]
[tex]\[
z^{-1} x^{-1} = \frac{1}{zx}
\][/tex]
So, the denominator becomes:
[tex]\[
x^{-1} y^{-1} + y^{-1} z^{-1} + z^{-1} x^{-1} = \frac{1}{xy} + \frac{1}{yz} + \frac{1}{zx}
\][/tex]
### Step 2: Combine the Denominator Terms
To combine these fractions, we find a common denominator, which is [tex]\( xyz \)[/tex]:
[tex]\[
\frac{1}{xy} = \frac{z}{xyz}
\][/tex]
[tex]\[
\frac{1}{yz} = \frac{x}{xyz}
\][/tex]
[tex]\[
\frac{1}{zx} = \frac{y}{xyz}
\][/tex]
Adding these fractions together, we get:
[tex]\[
\frac{1}{xy} + \frac{1}{yz} + \frac{1}{zx} = \frac{z}{xyz} + \frac{x}{xyz} + \frac{y}{xyz} = \frac{x + y + z}{xyz}
\][/tex]
Therefore, the simplified form of the denominator is:
[tex]\[
\frac{x + y + z}{xyz}
\][/tex]
### Step 3: Substitute the Simplified Denominator
Now, substitute the simplified denominator back into the original equation:
[tex]\[
\frac{x + y + z}{\frac{x + y + z}{xyz}} = xyz
\][/tex]
### Step 4: Simplify the Left Side
To simplify the left side, we use the fact that dividing by a fraction is the same as multiplying by its reciprocal:
[tex]\[
\frac{x + y + z}{\frac{x + y + z}{xyz}} = (x + y + z) \cdot \frac{xyz}{x + y + z} = xyz
\][/tex]
When we multiply [tex]\((x + y + z)\)[/tex] by [tex]\(\frac{xyz}{x + y + z}\)[/tex], the [tex]\((x + y + z)\)[/tex] terms in the numerator and the denominator cancel each other out:
[tex]\[
xyz
\][/tex]
So, the left side simplifies to [tex]\(xyz\)[/tex], which is identical to the right side:
[tex]\[
xyz = xyz
\][/tex]
### Conclusion
The equation [tex]\(\frac{x + y + z}{x^{-1} y^{-1} + y^{-1} z^{-1} + z^{-1} x^{-1}} = xyz\)[/tex] holds true as both sides simplify to the same expression [tex]\(xyz\)[/tex].
Thus, our step-by-step solution confirms that the given equation is indeed correct.
