Simplify square root of (x^4+3)^12

$\sqrt{{(x^{4} + 3)}^{12}}$

Step 1

Rewrite ${(x^{4} + 3)}^{12}$ as ${({(x^{4} + 3)}^{6})}^{2}$ .

$\sqrt{{({(x^{4} + 3)}^{6})}^{2}}$

Step 2

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

${(x^{4} + 3)}^{6}$

Step 3

Use the Binomial Theorem.

${(x^{4})}^{6} + 6 {(x^{4})}^{5} \cdot 3 + 15 {(x^{4})}^{4} \cdot 3^{2} + 20 {(x^{4})}^{3} \cdot 3^{3} + 15 {(x^{4})}^{2} \cdot 3^{4} + 6 x^{4} \cdot 3^{5} + 3^{6}$

Step 4

Simplify each term.

Multiply the exponents in ${(x^{4})}^{6}$ .

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

$x^{4 \cdot 6} + 6 {(x^{4})}^{5} \cdot 3 + 15 {(x^{4})}^{4} \cdot 3^{2} + 20 {(x^{4})}^{3} \cdot 3^{3} + 15 {(x^{4})}^{2} \cdot 3^{4} + 6 x^{4} \cdot 3^{5} + 3^{6}$

Multiply $4$ by $6$ .

$x^{24} + 6 {(x^{4})}^{5} \cdot 3 + 15 {(x^{4})}^{4} \cdot 3^{2} + 20 {(x^{4})}^{3} \cdot 3^{3} + 15 {(x^{4})}^{2} \cdot 3^{4} + 6 x^{4} \cdot 3^{5} + 3^{6}$

Multiply the exponents in ${(x^{4})}^{5}$ .

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

$x^{24} + 6 x^{4 \cdot 5} \cdot 3 + 15 {(x^{4})}^{4} \cdot 3^{2} + 20 {(x^{4})}^{3} \cdot 3^{3} + 15 {(x^{4})}^{2} \cdot 3^{4} + 6 x^{4} \cdot 3^{5} + 3^{6}$

Multiply $4$ by $5$ .

$x^{24} + 6 x^{20} \cdot 3 + 15 {(x^{4})}^{4} \cdot 3^{2} + 20 {(x^{4})}^{3} \cdot 3^{3} + 15 {(x^{4})}^{2} \cdot 3^{4} + 6 x^{4} \cdot 3^{5} + 3^{6}$

Multiply $3$ by $6$ .

$x^{24} + 18 x^{20} + 15 {(x^{4})}^{4} \cdot 3^{2} + 20 {(x^{4})}^{3} \cdot 3^{3} + 15 {(x^{4})}^{2} \cdot 3^{4} + 6 x^{4} \cdot 3^{5} + 3^{6}$

Multiply the exponents in ${(x^{4})}^{4}$ .

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

$x^{24} + 18 x^{20} + 15 x^{4 \cdot 4} \cdot 3^{2} + 20 {(x^{4})}^{3} \cdot 3^{3} + 15 {(x^{4})}^{2} \cdot 3^{4} + 6 x^{4} \cdot 3^{5} + 3^{6}$

Multiply $4$ by $4$ .

$x^{24} + 18 x^{20} + 15 x^{16} \cdot 3^{2} + 20 {(x^{4})}^{3} \cdot 3^{3} + 15 {(x^{4})}^{2} \cdot 3^{4} + 6 x^{4} \cdot 3^{5} + 3^{6}$

Raise $3$ to the power of $2$ .

$x^{24} + 18 x^{20} + 15 x^{16} \cdot 9 + 20 {(x^{4})}^{3} \cdot 3^{3} + 15 {(x^{4})}^{2} \cdot 3^{4} + 6 x^{4} \cdot 3^{5} + 3^{6}$

Multiply $9$ by $15$ .

$x^{24} + 18 x^{20} + 135 x^{16} + 20 {(x^{4})}^{3} \cdot 3^{3} + 15 {(x^{4})}^{2} \cdot 3^{4} + 6 x^{4} \cdot 3^{5} + 3^{6}$

Multiply the exponents in ${(x^{4})}^{3}$ .

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

$x^{24} + 18 x^{20} + 135 x^{16} + 20 x^{4 \cdot 3} \cdot 3^{3} + 15 {(x^{4})}^{2} \cdot 3^{4} + 6 x^{4} \cdot 3^{5} + 3^{6}$

Multiply $4$ by $3$ .

$x^{24} + 18 x^{20} + 135 x^{16} + 20 x^{12} \cdot 3^{3} + 15 {(x^{4})}^{2} \cdot 3^{4} + 6 x^{4} \cdot 3^{5} + 3^{6}$

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

$x^{24} + 18 x^{20} + 135 x^{16} + 20 x^{12} \cdot 27 + 15 {(x^{4})}^{2} \cdot 3^{4} + 6 x^{4} \cdot 3^{5} + 3^{6}$

Multiply $27$ by $20$ .

$x^{24} + 18 x^{20} + 135 x^{16} + 540 x^{12} + 15 {(x^{4})}^{2} \cdot 3^{4} + 6 x^{4} \cdot 3^{5} + 3^{6}$

Multiply the exponents in ${(x^{4})}^{2}$ .

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

$x^{24} + 18 x^{20} + 135 x^{16} + 540 x^{12} + 15 x^{4 \cdot 2} \cdot 3^{4} + 6 x^{4} \cdot 3^{5} + 3^{6}$

Multiply $4$ by $2$ .

$x^{24} + 18 x^{20} + 135 x^{16} + 540 x^{12} + 15 x^{8} \cdot 3^{4} + 6 x^{4} \cdot 3^{5} + 3^{6}$

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

$x^{24} + 18 x^{20} + 135 x^{16} + 540 x^{12} + 15 x^{8} \cdot 81 + 6 x^{4} \cdot 3^{5} + 3^{6}$

Multiply $81$ by $15$ .

$x^{24} + 18 x^{20} + 135 x^{16} + 540 x^{12} + 1215 x^{8} + 6 x^{4} \cdot 3^{5} + 3^{6}$

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

$x^{24} + 18 x^{20} + 135 x^{16} + 540 x^{12} + 1215 x^{8} + 6 x^{4} \cdot 243 + 3^{6}$

Multiply $243$ by $6$ .

$x^{24} + 18 x^{20} + 135 x^{16} + 540 x^{12} + 1215 x^{8} + 1458 x^{4} + 3^{6}$

Raise $3$ to the power of $6$ .

$x^{24} + 18 x^{20} + 135 x^{16} + 540 x^{12} + 1215 x^{8} + 1458 x^{4} + 729$

Simplify square root of (x^4+3)^12

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