Evaluate square root of 144(x+9)^10

$\sqrt{144 {(x + 9)}^{10}}$

Step 1

Rewrite $144 {(x + 9)}^{10}$ as ${(12 {(x + 9)}^{5})}^{2}$ .

$\sqrt{{(12 {(x + 9)}^{5})}^{2}}$

Step 2

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

$12 {(x + 9)}^{5}$

Step 3

Use the Binomial Theorem.

$12 (x^{5} + 5 x^{4} \cdot 9 + 10 x^{3} \cdot 9^{2} + 10 x^{2} \cdot 9^{3} + 5 x \cdot 9^{4} + 9^{5})$

Step 4

Simplify terms.

Simplify each term.

Multiply $9$ by $5$ .

$12 (x^{5} + 45 x^{4} + 10 x^{3} \cdot 9^{2} + 10 x^{2} \cdot 9^{3} + 5 x \cdot 9^{4} + 9^{5})$

Raise $9$ to the power of $2$ .

$12 (x^{5} + 45 x^{4} + 10 x^{3} \cdot 81 + 10 x^{2} \cdot 9^{3} + 5 x \cdot 9^{4} + 9^{5})$

Multiply $81$ by $10$ .

$12 (x^{5} + 45 x^{4} + 810 x^{3} + 10 x^{2} \cdot 9^{3} + 5 x \cdot 9^{4} + 9^{5})$

Raise $9$ to the power of $3$ .

$12 (x^{5} + 45 x^{4} + 810 x^{3} + 10 x^{2} \cdot 729 + 5 x \cdot 9^{4} + 9^{5})$

Multiply $729$ by $10$ .

$12 (x^{5} + 45 x^{4} + 810 x^{3} + 7290 x^{2} + 5 x \cdot 9^{4} + 9^{5})$

Raise $9$ to the power of $4$ .

$12 (x^{5} + 45 x^{4} + 810 x^{3} + 7290 x^{2} + 5 x \cdot 6561 + 9^{5})$

Multiply $6561$ by $5$ .

$12 (x^{5} + 45 x^{4} + 810 x^{3} + 7290 x^{2} + 32805 x + 9^{5})$

Raise $9$ to the power of $5$ .

$12 (x^{5} + 45 x^{4} + 810 x^{3} + 7290 x^{2} + 32805 x + 59049)$

Apply the distributive property.

$12 x^{5} + 12 (45 x^{4}) + 12 (810 x^{3}) + 12 (7290 x^{2}) + 12 (32805 x) + 12 \cdot 59049$

Step 5

Simplify.

Multiply $45$ by $12$ .

$12 x^{5} + 540 x^{4} + 12 (810 x^{3}) + 12 (7290 x^{2}) + 12 (32805 x) + 12 \cdot 59049$

Multiply $810$ by $12$ .

$12 x^{5} + 540 x^{4} + 9720 x^{3} + 12 (7290 x^{2}) + 12 (32805 x) + 12 \cdot 59049$

Multiply $7290$ by $12$ .

$12 x^{5} + 540 x^{4} + 9720 x^{3} + 87480 x^{2} + 12 (32805 x) + 12 \cdot 59049$

Multiply $32805$ by $12$ .

$12 x^{5} + 540 x^{4} + 9720 x^{3} + 87480 x^{2} + 393660 x + 12 \cdot 59049$

Multiply $12$ by $59049$ .

$12 x^{5} + 540 x^{4} + 9720 x^{3} + 87480 x^{2} + 393660 x + 708588$