Simplify square root of (x^2+6)^16

$\sqrt{{(x^{2} + 6)}^{16}}$

Step 1

Rewrite ${(x^{2} + 6)}^{16}$ as ${({(x^{2} + 6)}^{8})}^{2}$ .

$\sqrt{{({(x^{2} + 6)}^{8})}^{2}}$

Step 2

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

${(x^{2} + 6)}^{8}$

Step 3

Use the Binomial Theorem.

${(x^{2})}^{8} + 8 {(x^{2})}^{7} \cdot 6 + 28 {(x^{2})}^{6} \cdot 6^{2} + 56 {(x^{2})}^{5} \cdot 6^{3} + 70 {(x^{2})}^{4} \cdot 6^{4} + 56 {(x^{2})}^{3} \cdot 6^{5} + 28 {(x^{2})}^{2} \cdot 6^{6} + 8 x^{2} \cdot 6^{7} + 6^{8}$

Step 4

Simplify each term.

Multiply the exponents in ${(x^{2})}^{8}$ .

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

$x^{2 \cdot 8} + 8 {(x^{2})}^{7} \cdot 6 + 28 {(x^{2})}^{6} \cdot 6^{2} + 56 {(x^{2})}^{5} \cdot 6^{3} + 70 {(x^{2})}^{4} \cdot 6^{4} + 56 {(x^{2})}^{3} \cdot 6^{5} + 28 {(x^{2})}^{2} \cdot 6^{6} + 8 x^{2} \cdot 6^{7} + 6^{8}$

Multiply $2$ by $8$ .

$x^{16} + 8 {(x^{2})}^{7} \cdot 6 + 28 {(x^{2})}^{6} \cdot 6^{2} + 56 {(x^{2})}^{5} \cdot 6^{3} + 70 {(x^{2})}^{4} \cdot 6^{4} + 56 {(x^{2})}^{3} \cdot 6^{5} + 28 {(x^{2})}^{2} \cdot 6^{6} + 8 x^{2} \cdot 6^{7} + 6^{8}$

Multiply the exponents in ${(x^{2})}^{7}$ .

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

$x^{16} + 8 x^{2 \cdot 7} \cdot 6 + 28 {(x^{2})}^{6} \cdot 6^{2} + 56 {(x^{2})}^{5} \cdot 6^{3} + 70 {(x^{2})}^{4} \cdot 6^{4} + 56 {(x^{2})}^{3} \cdot 6^{5} + 28 {(x^{2})}^{2} \cdot 6^{6} + 8 x^{2} \cdot 6^{7} + 6^{8}$

Multiply $2$ by $7$ .

$x^{16} + 8 x^{14} \cdot 6 + 28 {(x^{2})}^{6} \cdot 6^{2} + 56 {(x^{2})}^{5} \cdot 6^{3} + 70 {(x^{2})}^{4} \cdot 6^{4} + 56 {(x^{2})}^{3} \cdot 6^{5} + 28 {(x^{2})}^{2} \cdot 6^{6} + 8 x^{2} \cdot 6^{7} + 6^{8}$

Multiply $6$ by $8$ .

$x^{16} + 48 x^{14} + 28 {(x^{2})}^{6} \cdot 6^{2} + 56 {(x^{2})}^{5} \cdot 6^{3} + 70 {(x^{2})}^{4} \cdot 6^{4} + 56 {(x^{2})}^{3} \cdot 6^{5} + 28 {(x^{2})}^{2} \cdot 6^{6} + 8 x^{2} \cdot 6^{7} + 6^{8}$

Multiply the exponents in ${(x^{2})}^{6}$ .

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

$x^{16} + 48 x^{14} + 28 x^{2 \cdot 6} \cdot 6^{2} + 56 {(x^{2})}^{5} \cdot 6^{3} + 70 {(x^{2})}^{4} \cdot 6^{4} + 56 {(x^{2})}^{3} \cdot 6^{5} + 28 {(x^{2})}^{2} \cdot 6^{6} + 8 x^{2} \cdot 6^{7} + 6^{8}$

Multiply $2$ by $6$ .

$x^{16} + 48 x^{14} + 28 x^{12} \cdot 6^{2} + 56 {(x^{2})}^{5} \cdot 6^{3} + 70 {(x^{2})}^{4} \cdot 6^{4} + 56 {(x^{2})}^{3} \cdot 6^{5} + 28 {(x^{2})}^{2} \cdot 6^{6} + 8 x^{2} \cdot 6^{7} + 6^{8}$

Raise $6$ to the power of $2$ .

$x^{16} + 48 x^{14} + 28 x^{12} \cdot 36 + 56 {(x^{2})}^{5} \cdot 6^{3} + 70 {(x^{2})}^{4} \cdot 6^{4} + 56 {(x^{2})}^{3} \cdot 6^{5} + 28 {(x^{2})}^{2} \cdot 6^{6} + 8 x^{2} \cdot 6^{7} + 6^{8}$

Multiply $36$ by $28$ .

$x^{16} + 48 x^{14} + 1008 x^{12} + 56 {(x^{2})}^{5} \cdot 6^{3} + 70 {(x^{2})}^{4} \cdot 6^{4} + 56 {(x^{2})}^{3} \cdot 6^{5} + 28 {(x^{2})}^{2} \cdot 6^{6} + 8 x^{2} \cdot 6^{7} + 6^{8}$

Multiply the exponents in ${(x^{2})}^{5}$ .

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

$x^{16} + 48 x^{14} + 1008 x^{12} + 56 x^{2 \cdot 5} \cdot 6^{3} + 70 {(x^{2})}^{4} \cdot 6^{4} + 56 {(x^{2})}^{3} \cdot 6^{5} + 28 {(x^{2})}^{2} \cdot 6^{6} + 8 x^{2} \cdot 6^{7} + 6^{8}$

Multiply $2$ by $5$ .

$x^{16} + 48 x^{14} + 1008 x^{12} + 56 x^{10} \cdot 6^{3} + 70 {(x^{2})}^{4} \cdot 6^{4} + 56 {(x^{2})}^{3} \cdot 6^{5} + 28 {(x^{2})}^{2} \cdot 6^{6} + 8 x^{2} \cdot 6^{7} + 6^{8}$

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

$x^{16} + 48 x^{14} + 1008 x^{12} + 56 x^{10} \cdot 216 + 70 {(x^{2})}^{4} \cdot 6^{4} + 56 {(x^{2})}^{3} \cdot 6^{5} + 28 {(x^{2})}^{2} \cdot 6^{6} + 8 x^{2} \cdot 6^{7} + 6^{8}$

Multiply $216$ by $56$ .

$x^{16} + 48 x^{14} + 1008 x^{12} + 12096 x^{10} + 70 {(x^{2})}^{4} \cdot 6^{4} + 56 {(x^{2})}^{3} \cdot 6^{5} + 28 {(x^{2})}^{2} \cdot 6^{6} + 8 x^{2} \cdot 6^{7} + 6^{8}$

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

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

$x^{16} + 48 x^{14} + 1008 x^{12} + 12096 x^{10} + 70 x^{2 \cdot 4} \cdot 6^{4} + 56 {(x^{2})}^{3} \cdot 6^{5} + 28 {(x^{2})}^{2} \cdot 6^{6} + 8 x^{2} \cdot 6^{7} + 6^{8}$

Multiply $2$ by $4$ .

$x^{16} + 48 x^{14} + 1008 x^{12} + 12096 x^{10} + 70 x^{8} \cdot 6^{4} + 56 {(x^{2})}^{3} \cdot 6^{5} + 28 {(x^{2})}^{2} \cdot 6^{6} + 8 x^{2} \cdot 6^{7} + 6^{8}$

Raise $6$ to the power of $4$ .

$x^{16} + 48 x^{14} + 1008 x^{12} + 12096 x^{10} + 70 x^{8} \cdot 1296 + 56 {(x^{2})}^{3} \cdot 6^{5} + 28 {(x^{2})}^{2} \cdot 6^{6} + 8 x^{2} \cdot 6^{7} + 6^{8}$

Multiply $1296$ by $70$ .

$x^{16} + 48 x^{14} + 1008 x^{12} + 12096 x^{10} + 90720 x^{8} + 56 {(x^{2})}^{3} \cdot 6^{5} + 28 {(x^{2})}^{2} \cdot 6^{6} + 8 x^{2} \cdot 6^{7} + 6^{8}$

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

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

$x^{16} + 48 x^{14} + 1008 x^{12} + 12096 x^{10} + 90720 x^{8} + 56 x^{2 \cdot 3} \cdot 6^{5} + 28 {(x^{2})}^{2} \cdot 6^{6} + 8 x^{2} \cdot 6^{7} + 6^{8}$

Multiply $2$ by $3$ .

$x^{16} + 48 x^{14} + 1008 x^{12} + 12096 x^{10} + 90720 x^{8} + 56 x^{6} \cdot 6^{5} + 28 {(x^{2})}^{2} \cdot 6^{6} + 8 x^{2} \cdot 6^{7} + 6^{8}$

Raise $6$ to the power of $5$ .

$x^{16} + 48 x^{14} + 1008 x^{12} + 12096 x^{10} + 90720 x^{8} + 56 x^{6} \cdot 7776 + 28 {(x^{2})}^{2} \cdot 6^{6} + 8 x^{2} \cdot 6^{7} + 6^{8}$

Multiply $7776$ by $56$ .

$x^{16} + 48 x^{14} + 1008 x^{12} + 12096 x^{10} + 90720 x^{8} + 435456 x^{6} + 28 {(x^{2})}^{2} \cdot 6^{6} + 8 x^{2} \cdot 6^{7} + 6^{8}$

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

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

$x^{16} + 48 x^{14} + 1008 x^{12} + 12096 x^{10} + 90720 x^{8} + 435456 x^{6} + 28 x^{2 \cdot 2} \cdot 6^{6} + 8 x^{2} \cdot 6^{7} + 6^{8}$

Multiply $2$ by $2$ .

$x^{16} + 48 x^{14} + 1008 x^{12} + 12096 x^{10} + 90720 x^{8} + 435456 x^{6} + 28 x^{4} \cdot 6^{6} + 8 x^{2} \cdot 6^{7} + 6^{8}$

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

$x^{16} + 48 x^{14} + 1008 x^{12} + 12096 x^{10} + 90720 x^{8} + 435456 x^{6} + 28 x^{4} \cdot 46656 + 8 x^{2} \cdot 6^{7} + 6^{8}$

Multiply $46656$ by $28$ .

$x^{16} + 48 x^{14} + 1008 x^{12} + 12096 x^{10} + 90720 x^{8} + 435456 x^{6} + 1306368 x^{4} + 8 x^{2} \cdot 6^{7} + 6^{8}$

Raise $6$ to the power of $7$ .

$x^{16} + 48 x^{14} + 1008 x^{12} + 12096 x^{10} + 90720 x^{8} + 435456 x^{6} + 1306368 x^{4} + 8 x^{2} \cdot 279936 + 6^{8}$

Multiply $279936$ by $8$ .

$x^{16} + 48 x^{14} + 1008 x^{12} + 12096 x^{10} + 90720 x^{8} + 435456 x^{6} + 1306368 x^{4} + 2239488 x^{2} + 6^{8}$

Raise $6$ to the power of $8$ .

$x^{16} + 48 x^{14} + 1008 x^{12} + 12096 x^{10} + 90720 x^{8} + 435456 x^{6} + 1306368 x^{4} + 2239488 x^{2} + 1679616$

Simplify square root of (x^2+6)^16

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