Simplify ( square root of x-5)^7

${(\sqrt{x - 5})}^{7}$

Step 1

Rewrite ${\sqrt{x - 5}}^{7}$ as $\sqrt{{(x - 5)}^{7}}$ .

$\sqrt{{(x - 5)}^{7}}$

Step 2

Rewrite ${(x - 5)}^{7}$ as ${({(x - 5)}^{3})}^{2} (x - 5)$ .

Factor out ${(x - 5)}^{6}$ .

$\sqrt{{(x - 5)}^{6} (x - 5)}$

Rewrite ${(x - 5)}^{6}$ as ${({(x - 5)}^{3})}^{2}$ .

$\sqrt{{({(x - 5)}^{3})}^{2} (x - 5)}$

Step 3

Pull terms out from under the radical.

${(x - 5)}^{3} \sqrt{x - 5}$

Step 4

Use the Binomial Theorem.

$(x^{3} + 3 x^{2} \cdot - 5 + 3 x {(- 5)}^{2} + {(- 5)}^{3}) \sqrt{x - 5}$

Step 5

Simplify terms.

Simplify each term.

Multiply $- 5$ by $3$ .

$(x^{3} - 15 x^{2} + 3 x {(- 5)}^{2} + {(- 5)}^{3}) \sqrt{x - 5}$

Raise $- 5$ to the power of $2$ .

$(x^{3} - 15 x^{2} + 3 x \cdot 25 + {(- 5)}^{3}) \sqrt{x - 5}$

Multiply $25$ by $3$ .

$(x^{3} - 15 x^{2} + 75 x + {(- 5)}^{3}) \sqrt{x - 5}$

Raise $- 5$ to the power of $3$ .

$(x^{3} - 15 x^{2} + 75 x - 125) \sqrt{x - 5}$

Apply the distributive property.

$x^{3} \sqrt{x - 5} - 15 x^{2} \sqrt{x - 5} + 75 x \sqrt{x - 5} - 125 \sqrt{x - 5}$