Evaluate square root of 49(x^3+4)^6

$\sqrt{49 {(x^{3} + 4)}^{6}}$

Step 1

Rewrite $49 {(x^{3} + 4)}^{6}$ as ${(7 {(x^{3} + 4)}^{3})}^{2}$ .

$\sqrt{{(7 {(x^{3} + 4)}^{3})}^{2}}$

Step 2

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

$7 {(x^{3} + 4)}^{3}$

Step 3

Use the Binomial Theorem.

$7 ({(x^{3})}^{3} + 3 {(x^{3})}^{2} \cdot 4 + 3 x^{3} \cdot 4^{2} + 4^{3})$

Step 4

Simplify terms.

Simplify each term.

Multiply the exponents in ${(x^{3})}^{3}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$7 (x^{3 \cdot 3} + 3 {(x^{3})}^{2} \cdot 4 + 3 x^{3} \cdot 4^{2} + 4^{3})$

Multiply $3$ by $3$ .

$7 (x^{9} + 3 {(x^{3})}^{2} \cdot 4 + 3 x^{3} \cdot 4^{2} + 4^{3})$

Multiply the exponents in ${(x^{3})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$7 (x^{9} + 3 x^{3 \cdot 2} \cdot 4 + 3 x^{3} \cdot 4^{2} + 4^{3})$

Multiply $3$ by $2$ .

$7 (x^{9} + 3 x^{6} \cdot 4 + 3 x^{3} \cdot 4^{2} + 4^{3})$

Multiply $4$ by $3$ .

$7 (x^{9} + 12 x^{6} + 3 x^{3} \cdot 4^{2} + 4^{3})$

Raise $4$ to the power of $2$ .

$7 (x^{9} + 12 x^{6} + 3 x^{3} \cdot 16 + 4^{3})$

Multiply $16$ by $3$ .

$7 (x^{9} + 12 x^{6} + 48 x^{3} + 4^{3})$

Raise $4$ to the power of $3$ .

$7 (x^{9} + 12 x^{6} + 48 x^{3} + 64)$

Apply the distributive property.

$7 x^{9} + 7 (12 x^{6}) + 7 (48 x^{3}) + 7 \cdot 64$

Step 5

Simplify.

Multiply $12$ by $7$ .

$7 x^{9} + 84 x^{6} + 7 (48 x^{3}) + 7 \cdot 64$

Multiply $48$ by $7$ .

$7 x^{9} + 84 x^{6} + 336 x^{3} + 7 \cdot 64$

Multiply $7$ by $64$ .

$7 x^{9} + 84 x^{6} + 336 x^{3} + 448$