Question

question
select the equivalent expression.
\\(\frac{xy^{-2}y^{-8}}{x^4}\\)
answer
\\(\frac{1}{x^3}\\) \\(x^3\\)
\\(x^3y^{10}\\) \\(\frac{1}{x^3y^{10}}\\)

Explanation:

Step1: Simplify y - terms

Using the rule \(a^m \cdot a^n = a^{m + n}\), for \(y^{-2} \cdot y^{-8}\), we have \(y^{-2 + (-8)} = y^{-10}\). So the numerator becomes \(x \cdot y^{-10}\).

Step2: Simplify x - terms

Now we have \(\frac{x \cdot y^{-10}}{x^4}\). Using the rule \(\frac{a^m}{a^n}=a^{m - n}\) for the x - terms (\(x^1\) and \(x^4\)), we get \(x^{1 - 4}=x^{-3}\).

Step3: Rewrite negative exponents

Using the rule \(a^{-n}=\frac{1}{a^n}\), \(x^{-3}y^{-10}=\frac{1}{x^{3}y^{10}}\).

Answer:

\(\frac{1}{x^{3}y^{10}}\) (the last option: \(\boldsymbol{\frac{1}{x^{3}y^{10}}}\))