Part 1Use any method to simplify the complex fraction.StartStartFraction StartFraction 25 z Superscript 4 Baseline x Superscript 4 Over 49 EndFraction OverOver StartFraction 175 x squared z Superscript 6 Over 21 EndFraction EndEndFraction 25z4x449 175x2z621 Part 1The fraction simplifies to [input]enter your response here .

Question

BlackTom AI · Accepted Answer

Divide the two rational expressions by multiplying by the reciprocal of the denominator: (25 z^4 x^4 / 49) ÷ (175 x^2 z^6 / 21) = (25 z^4 x^4 / 49) * (21 / 175 x^2 z^6).

1) Relevant knowledge points and concepts
- Division of rational expressions: A/B ÷ C/D = (A/B) * (D/C).
- Exponent rules: x^a / x^b = x^(a−b).
- Simplifying numerical coefficients by factoring and canceling common factors.

2) Step-by-step reasoning and intermediate calculations
Start with the reciprocal multiplication:
(25 z^4 x^4 / 49) * (21 / (175 x^2 z^6))

Separate numerical and variable parts:
Numerical part: (25 * 21) / (49 * 175)
x-part: x^4 / x^2 = x^(4−2) = x^2
z-part: z^4 / z^6 = z^(4−6) = z^(−2) = 1/z^2

Simplify the numerical fraction:
- 25/175 = 1/7 (since 25·7 = 175)
- 21/49 = 3/7 (since 21 = 3·7 and 49 = 7·7)
Multiply these simplified ratios: (1/7) * (3/7) = 3/49

Alternatively: (25*21) = 525 and (49*175) = 8575; 525/8575 reduces to 3/49 after dividing numerator and denominator by 175.

Combine everything:
Numerical: 3/49
Variables: x^2 * 1/z^2 = x^2 / z^2

So the simplified expression is 3x^2/(49 z^2).

3) Why this answer is correct
The algebraic steps follow the rules for dividing fractions (multiply by reciprocal), exponent subtraction for like bases, and cancellation of common numerical factors. Each cancellation is valid and reduces the expression to lowest terms.

4) Why common incorrect answers would be wrong
- Forgetting to invert the denominator yields a wrong operation.
- Subtracting exponents in the wrong order could produce z^2 instead of z^(−2).
- Failing to reduce 25/175 and 21/49 leads to a non-simplified numeric factor.

5) Conclusion
The fully simplified result is 3x^2/(49 z^2), which matches the provided correct answer.

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