Question

question
select the equivalent expression.
\\(\frac{x^{-2}y^{-8}}{x^{8}y^{7}}\\)

answer
\\(\circ\\) \\(\frac{1}{x^{10}y^{15}}\\) \\(\circ\\) \\(\frac{1}{x^{6}y^{15}}\\) \\(\circ\\) \\(x^{6}y^{15}\\) \\(\circ\\) \\(x^{10}y^{15}\\)

Explanation:

Step1: Apply exponent rule for division

When dividing exponents with the same base, we subtract the exponents: \( \frac{a^m}{a^n} = a^{m - n} \). For the given expression \( \frac{x^{-2}y^{-8}}{x^8y^7} \), we can split it into the quotient of \( x \)-terms and \( y \)-terms: \( \frac{x^{-2}}{x^8} \cdot \frac{y^{-8}}{y^7} \).

Step2: Simplify \( x \)-terms

For the \( x \)-terms, using the exponent rule: \( x^{-2 - 8} = x^{-10} \).

Step3: Simplify \( y \)-terms

For the \( y \)-terms, using the exponent rule: \( y^{-8 - 7} = y^{-15} \).

Step4: Apply negative exponent rule

A negative exponent means the reciprocal: \( a^{-n} = \frac{1}{a^n} \). So \( x^{-10}y^{-15} = \frac{1}{x^{10}y^{15}} \).

Answer:

\( \frac{1}{x^{10}y^{15}} \) (corresponding to the first option in the answer choices)