Question

which expression is equivalent to \\(\frac{3x^{-6}y^{-3}}{15x^{2}y^{10}}\\)? assume \\(x \
eq 0, y \
eq 0\\). \\(\frac{1}{5x^{8}y^{13}}\\) \\(\frac{5}{x^{4}y^{7}}\\) \\(\frac{y^{7}}{5x^{4}}\\) \\(\frac{5}{x^{12}y^{30}}\\)

Explanation:

Step1: Simplify the coefficient

Simplify the fraction of the coefficients: $\frac{3}{15} = \frac{1}{5}$.

Step2: Simplify the \(x\)-terms

Use the rule of exponents \(a^m \div a^n = a^{m - n}\) for \(x\)-terms: \(x^{-6} \div x^{2} = x^{-6 - 2} = x^{-8}=\frac{1}{x^{8}}\).

Step3: Simplify the \(y\)-terms

Use the rule of exponents for \(y\)-terms: \(y^{-3} \div y^{10} = y^{-3 - 10} = y^{-13}=\frac{1}{y^{13}}\).

Step4: Combine all parts

Multiply the simplified coefficient, \(x\)-term, and \(y\)-term together: \(\frac{1}{5} \times \frac{1}{x^{8}} \times \frac{1}{y^{13}}=\frac{1}{5x^{8}y^{13}}\).

Answer:

\(\frac{1}{5x^{8}y^{13}}\) (the first option)