show that (7pqr-5)²-(7pqr +5)² /(pqr + 1)² - (pqr-1)² =-35

To solve the given expression, let's simplify it step-by-step. The expression is:

\[
\frac{(7pqr - 5)^2 - (7pqr + 5)^2}{(pqr + 1)^2 - (pqr - 1)^2}
\]

### Step 1: Expand the Numerator

The numerator is of the form \(a^2 - b^2\), which is a difference of squares. Recall that:

\[
a^2 - b^2 = (a - b)(a + b)
\]

Let \(a = 7pqr - 5\) and \(b = 7pqr + 5\). Then:

\[
(7pqr - 5)^2 - (7pqr + 5)^2 = ((7pqr - 5) - (7pqr + 5)) \cdot ((7pqr - 5) + (7pqr + 5))
\]

Simplify inside the parentheses:

1. \((7pqr - 5) - (7pqr + 5) = 7pqr - 5 - 7pqr - 5 = -10\)

2. \((7pqr - 5) + (7pqr + 5) = 7pqr - 5 + 7pqr + 5 = 14pqr\)

So, the numerator becomes:

\[
-10 \cdot 14pqr = -140pqr
\]

### Step 2: Expand the Denominator

The denominator is also a difference of squares. Let \(c = pqr + 1\) and \(d = pqr - 1\). Then:

\[
(pqr + 1)^2 - (pqr - 1)^2 = ((pqr + 1) - (pqr - 1)) \cdot ((pqr + 1) + (pqr - 1))
\]

Simplify inside the parentheses:

1. \((pqr + 1) - (pqr - 1) = pqr + 1 - pqr + 1 = 2\)

2. \((pqr + 1) + (pqr - 1) = pqr + 1 + pqr - 1 = 2pqr\)

So, the denominator becomes:

\[
2 \cdot 2pqr = 4pqr
\]

### Step 3: Simplify the Expression

Substitute the simplified numerator and denominator back into the expression:

\[
\frac{-140pqr}{4pqr}
\]

Cancel out the common term \(pqr\) (assuming \(pqr \neq 0\)):

\[
\frac{-140}{4} = -35
\]

Thus, the given expression simplifies to:

\[
-35
\]

### Conclusion

We have shown that:

\[
\frac{(7pqr - 5)^2 - (7pqr + 5)^2}{(pqr + 1)^2 - (pqr - 1)^2} = -35
\]

This completes the proof.

Would you like further details or have any questions?

### Related Questions

1. How do you simplify expressions involving differences of squares?
2. Can you show a similar proof with different coefficients?
3. What happens if \(pqr = 0\) in this expression?
4. How do you derive the formula for the difference of squares?
5. How would this problem change if the exponents were different?

**Tip:** Remember that simplifying complex fractions often involves recognizing patterns such as the difference of squares.

Learn how to simplify the expression (7pqr - 5)^2 - (7pqr + 5)^2 / (pqr + 1)^2 - (pqr - 1)^2 = -35 with detailed steps. This problem involves the difference of squares and quadratic equations. Suitable for high school and above.

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