(x^2+6x+9)/(x^2-6x+9)/((x^2+3x)/(x-3))^5

To simplify the expression \(\frac{x^2+6x+9}{x^2-6x+9}\div\left(\frac{x^2+3x}{x-3}\right)^5\), let's begin by factoring each component of the expression. 1. **Factor the numerators and denominators**: - The first term in the numerator: \[ x^2 + 6x + 9 = (x + 3)^2 \] - The first term in the denominator: \[ x^2 - 6x + 9 = (x - 3)^2 \] - For the term \(\frac{x^2 + 3x}{x - 3}\): \[ x^2 + 3x = x(x + 3) \] So, \[ \frac{x(x + 3)}{x - 3} \] 2. **Combine the expression**: Now rewrite the expression using the factored forms: \[ \frac{(x + 3)^2}{(x - 3)^2} \div \left(\frac{x(x + 3)}{x - 3}\right)^5 \] This can also be expressed as: \[ \frac{(x + 3)^2}{(x - 3)^2} \cdot \left(\frac{x - 3}{x(x + 3)}\right)^5 \] 3. **Combine the factors**: The entire expression simplifies to: \[ \frac{(x + 3)^2}{(x - 3)^2} \cdot \frac{(x - 3)^5}{(x(x + 3))^5} \] This results in: \[ \frac{(x + 3)^2 \cdot (x - 3)^5}{(x - 3)^2 \cdot (x(x + 3))^5} \] 4. **Cancel common terms**: Cancel \((x + 3)^2\) from the numerator and denominator: \[ \frac{(x - 3)^5}{(x - 3)^2 \cdot (x(x + 3))^5} \] This results in: \[ \frac{(x - 3)^{5-2}}{x^5(x + 3)^5} = \frac{(x - 3)^3}{x^5(x + 3)^5} \] Putting it all together, the simplified expression is: \[ \frac{(x - 3)^3}{x^5(x + 3)^5} \]