Question

divide and simplify.\\(
\frac{(x^{3}y)^{2}}{(x + 6y)^{2}}\div\frac{x^{2}y}{(x + 6y)^{3}}\\)
\\(\quad\\), \\(y\
eq0,\quad x\
eq0, - 6y\\)

Explanation:

Step1: Convert division to multiplication

Recall that dividing by a fraction is the same as multiplying by its reciprocal. So we rewrite the division as multiplication:
$$\frac{(x^3y)^2}{(x + 6y)^2} \times \frac{(x + 6y)^3}{x^2y}$$

Step2: Simplify the numerator of the first fraction

Use the power of a product rule \((ab)^n=a^n b^n\) and power of a power rule \((a^m)^n=a^{mn}\) on \((x^3y)^2\):
$$(x^3)^2y^2 = x^{6}y^2$$
So the expression becomes:
$$\frac{x^{6}y^2}{(x + 6y)^2} \times \frac{(x + 6y)^3}{x^2y}$$

Step3: Multiply the numerators and denominators

Multiply the numerators together and the denominators together:
$$\frac{x^{6}y^2 \times (x + 6y)^3}{(x + 6y)^2 \times x^2y}$$

Step4: Simplify using exponent rules

For the \(x\)-terms, use the quotient rule \(a^m\div a^n=a^{m - n}\): \(x^{6}\div x^{2}=x^{6 - 2}=x^{4}\)
For the \(y\)-terms: \(y^2\div y = y^{2 - 1}=y\)
For the \((x + 6y)\)-terms: \((x + 6y)^3\div(x + 6y)^2=(x + 6y)^{3 - 2}=(x + 6y)\)

Multiply the simplified terms together: \(x^{4}\times y\times(x + 6y)=x^{4}y(x + 6y)\)

Answer:

\(x^{4}y(x + 6y)\)