If y is a mean proportional between X and z prove that x square minus y square + z square divided by x to the power minus 2 minus 5 to the power minus 2 + Z2 the power minus 2 equal to y to the power 4

Given that \( y \) is a mean proportional between \( x \) and \( z \), we have the relationship:

\[
y^2 = xz
\]

We need to prove the expression:

\[
\frac{x^2 - y^2 + z^2}{x^{-2} - y^{-2} + z^{-2}} = y^4
\]

### Step 1: Simplify the Numerator

The numerator is:

\[
x^2 - y^2 + z^2
\]

Using the given relationship \( y^2 = xz \), substitute \( y^2 \) in the expression:

\[
x^2 - xz + z^2
\]

### Step 2: Simplify the Denominator

The denominator is:

\[
x^{-2} - y^{-2} + z^{-2}
\]

We can rewrite this as:

\[
\frac{1}{x^2} - \frac{1}{y^2} + \frac{1}{z^2}
\]

Substituting \( y^2 = xz \), the expression becomes:

\[
\frac{1}{x^2} - \frac{1}{xz} + \frac{1}{z^2}
\]

Simplify the terms:

\[
= \frac{z^2 - xz + x^2}{x^2 z^2}
\]

### Step 3: Divide the Simplified Numerator by the Denominator

The expression to simplify now is:

\[
\frac{x^2 - xz + z^2}{\frac{z^2 - xz + x^2}{x^2 z^2}}
\]

Multiplying by the reciprocal of the denominator:

\[
= (x^2 - xz + z^2) \times \frac{x^2 z^2}{z^2 - xz + x^2}
\]

This simplifies to:

\[
= x^2 z^2
\]

Using \( y^2 = xz \), we get:

\[
x^2 z^2 = (xz)^2 = y^4
\]

### Conclusion:

\[
\frac{x^2 - y^2 + z^2}{x^{-2} - y^{-2} + z^{-2}} = y^4
\]

Thus, the expression is proven.

Would you like more details or have any questions?

### Related Questions:

1. What is the geometric mean and how is it used in various applications?
2. Can you prove the relation between mean proportional and geometric sequences?
3. How does simplification of fractions involving negative exponents work?
4. How can we apply similar techniques in calculus or higher-level algebra?
5. What are other examples of mean proportional problems and their proofs?

### Tip:

Always look for substitutions that simplify terms, especially when working with complex expressions involving squares or exponents.

If y is a mean proportional between x and z, prove that (x^2 - y^2 + z^2) / (x^-2 - y^-2 + z^-2) = y^4.

Explore the proof of the expression where y is a mean proportional between x and z: (x^2 - y^2 + z^2) / (x^-2 - y^-2 + z^-2) = y^4. This problem delves into key algebraic concepts, including exponents, proportions, and algebraic manipulation, suitable for students in Grades 10-12.

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