Simplify (x^2+18x+81)/((x+9)^5 square root of x+9)

Rewrite ${\displaystyle 81}$ as ${\displaystyle 9^2}$.

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

Rewrite the polynomial.

Factor using the perfect square trinomial rule ${\displaystyle a^2+2ab+b^2={(a+b)}^2}$, where ${\displaystyle a=x}$ and ${\displaystyle b=9}$.

Multiply by ${\displaystyle 1}$.

Cancel the common factors.

Multiply ${\displaystyle \frac{1}{{(x+9)}^3\sqrt{x+9}}}$ by ${\displaystyle \frac{\sqrt{x+9}}{\sqrt{x+9}}}$.

Move ${\displaystyle \sqrt{x+9}}$.

Raise ${\displaystyle \sqrt{x+9}}$ to the power of ${\displaystyle 1}$.

Raise ${\displaystyle \sqrt{x+9}}$ to the power of ${\displaystyle 1}$.

Use the power rule ${\displaystyle a^m{a}^n=a^{m+n}}$ to combine exponents.

Add ${\displaystyle 1}$ and ${\displaystyle 1}$.

Rewrite ${\displaystyle {\sqrt{x+9}}^2}$ as ${\displaystyle x+9}$.

Multiply ${\displaystyle {(x+9)}^3}$ by ${\displaystyle x+9}$.

Add ${\displaystyle 3}$ and ${\displaystyle 1}$.