Question

simplify the expression. write the result using positive exponents only. \\(\left(\frac{x^{-2} y^{2}}{x^{2} y^{9}}\
ight)^{3}\\) (simplify your answer. use positive exponents only.) \\(\left(\frac{x^{-2} y^{2}}{x^{2} y^{9}}\
ight)^{3} = \square\\)

Explanation:

Step1: Simplify the fraction inside the parentheses

First, simplify the fraction \(\frac{x^{-2}y^{2}}{x^{2}y^{9}}\) using the quotient rule for exponents \( \frac{a^m}{a^n}=a^{m - n}\).
For the \(x\)-terms: \(x^{-2-2}=x^{-4}\)
For the \(y\)-terms: \(y^{2 - 9}=y^{-7}\)
So the fraction simplifies to \(x^{-4}y^{-7}\)

Step2: Apply the power of a power rule

Now, raise the result to the power of 3. The power of a power rule is \((a^m)^n=a^{m\times n}\)
For the \(x\)-term: \((x^{-4})^3 = x^{-4\times3}=x^{-12}\)
For the \(y\)-term: \((y^{-7})^3 = y^{-7\times3}=y^{-21}\)

Step3: Convert to positive exponents

Use the rule \(a^{-n}=\frac{1}{a^n}\) to convert negative exponents to positive.
\(x^{-12}=\frac{1}{x^{12}}\) and \(y^{-21}=\frac{1}{y^{21}}\), so the expression becomes \(\frac{1}{x^{12}y^{21}}\)

Answer:

\(\frac{1}{x^{12}y^{21}}\)