Question

simplify the expression.
\frac{(xy)^{6}}{y^{5}} \cdot \frac{4}{x^{7}}

Explanation:

Step1: Expand \((xy)^6\)

Using the power of a product rule \((ab)^n = a^n b^n\), we get \((xy)^6 = x^6 y^6\). So the expression becomes \(\frac{x^6 y^6}{y^5} \cdot \frac{4}{x^7}\).

Step2: Simplify \(\frac{y^6}{y^5}\)

Using the quotient rule for exponents \(\frac{a^m}{a^n}=a^{m - n}\), we have \(\frac{y^6}{y^5}=y^{6 - 5}=y\). Now the expression is \(\frac{x^6 \cdot y}{1} \cdot \frac{4}{x^7}\).

Step3: Simplify \(\frac{x^6}{x^7}\)

Using the quotient rule for exponents again, \(\frac{x^6}{x^7}=x^{6 - 7}=x^{-1}=\frac{1}{x}\). The expression now is \(x^6 y \cdot \frac{4}{x^7}=4y\cdot\frac{x^6}{x^7}\). Substituting \(\frac{x^6}{x^7}=\frac{1}{x}\), we get \(4y\cdot\frac{1}{x}=\frac{4y}{x}\).

Answer:

\(\frac{4y}{x}\)