Question

a45 - exponents power rule
score: 8/10 answered: 8/10
question 9
simplify the following expression completely. write your answer using only positive expre
\\(\frac{(x^4 y^2)^5 (x y^5)^{-2}}{x^4 y^3}\\) = \\(square\\)

Explanation:

Step1: Apply Power of a Product Rule

For \((x^4y^2)^5\), use \((ab)^n = a^n b^n\) and \((a^m)^n=a^{mn}\):
\((x^4)^5(y^2)^5 = x^{20}y^{10}\)

For \((xy^5)^{-2}\), use the same rules:
\(x^{-2}(y^5)^{-2}=x^{-2}y^{-10}\)

Now the numerator is \(x^{20}y^{10} \cdot x^{-2}y^{-10}\). Use \(a^m \cdot a^n = a^{m + n}\):
\(x^{20 + (-2)}y^{10 + (-10)} = x^{18}y^{0}\). Since \(y^0 = 1\) (for \(y
eq 0\)), the numerator simplifies to \(x^{18}\).

Step2: Divide by \(x^4y^3\)

The expression is now \(\frac{x^{18}}{x^4y^3}\). Use \(\frac{a^m}{a^n}=a^{m - n}\) for the \(x\)-terms:
\(x^{18 - 4}y^{0 - 3}=x^{14}y^{-3}\).

Step3: Convert Negative Exponent to Positive

Use \(a^{-n}=\frac{1}{a^n}\) for \(y^{-3}\):
\(x^{14} \cdot \frac{1}{y^3}=\frac{x^{14}}{y^3}\)

Answer:

\(\boldsymbol{\frac{x^{14}}{y^3}}\)