To simplify the given expression [tex]\(\frac{x^5 \cdot (y^{-8} \cdot z^{-7})^2}{x^8 \cdot y^{-1} \cdot z^{-2}}\)[/tex], let's follow these steps:
1. Simplify the Numerator:
- Start with [tex]\((y^{-8} \cdot z^{-7})^2\)[/tex].
- Apply the power of a product rule [tex]\((a \cdot b)^n = a^n \cdot b^n\)[/tex] to get [tex]\(y^{-8 \cdot 2} \cdot z^{-7 \cdot 2} = y^{-16} \cdot z^{-14}\)[/tex].
- Now, substitute back into the numerator: [tex]\(x^5 \cdot (y^{-16} \cdot z^{-14}) = x^5 \cdot y^{-16} \cdot z^{-14}\)[/tex].
2. Combine the Numerator and Denominator:
- We now have the expression [tex]\(\frac{x^5 \cdot y^{-16} \cdot z^{-14}}{x^8 \cdot y^{-1} \cdot z^{-2}}\)[/tex].
3. Simplify the Exponents:
- Use the rule for division of exponents [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]:
- For [tex]\(x\)[/tex]: [tex]\(x^{5-8} = x^{-3}\)[/tex].
- For [tex]\(y\)[/tex]: [tex]\(y^{-16 - (-1)} = y^{-16 + 1} = y^{-15}\)[/tex].
- For [tex]\(z\)[/tex]: [tex]\(z^{-14 - (-2)} = z^{-14 + 2} = z^{-12}\)[/tex].
4. Rewrite with Positive Exponents:
- We have [tex]\(x^{-3} \cdot y^{-15} \cdot z^{-12}\)[/tex].
- Rewrite with positive exponents using the rule [tex]\(a^{-n} = \frac{1}{a^n}\)[/tex]: [tex]\(\frac{1}{x^3 \cdot y^{15} \cdot z^{12}}\)[/tex].
So, the simplified expression is [tex]\(\frac{1}{x^3 \cdot y^{15} \cdot z^{12}}\)[/tex].
