Simplify cube root of 729(x-7)^6

$\sqrt[3]{729 {(x - 7)}^{6}}$

Step 1

Rewrite $729 {(x - 7)}^{6}$ as ${(9 {(x - 7)}^{2})}^{3}$ .

$\sqrt[3]{{(9 {(x - 7)}^{2})}^{3}}$

Step 2

Pull terms out from under the radical, assuming real numbers.

$9 {(x - 7)}^{2}$

Step 3

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

$9 ((x - 7) (x - 7))$

Step 4

Expand $(x - 7) (x - 7)$ using the FOIL Method.

Apply the distributive property.

$9 (x (x - 7) - 7 (x - 7))$

Apply the distributive property.

$9 (x \cdot x + x \cdot - 7 - 7 (x - 7))$

Apply the distributive property.

$9 (x \cdot x + x \cdot - 7 - 7 x - 7 \cdot - 7)$

Step 5

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$9 (x^{2} + x \cdot - 7 - 7 x - 7 \cdot - 7)$

Move $- 7$ to the left of $x$ .

$9 (x^{2} - 7 \cdot x - 7 x - 7 \cdot - 7)$

Multiply $- 7$ by $- 7$ .

$9 (x^{2} - 7 x - 7 x + 49)$

Subtract $7 x$ from $- 7 x$ .

$9 (x^{2} - 14 x + 49)$

Step 6

Apply the distributive property.

$9 x^{2} + 9 (- 14 x) + 9 \cdot 49$

Step 7

Simplify.

Multiply $- 14$ by $9$ .

$9 x^{2} - 126 x + 9 \cdot 49$

Multiply $9$ by $49$ .

$9 x^{2} - 126 x + 441$