Algebra Examples

$\sqrt[6]{{(x^{2} + 2)}^{18}}$

Step 1

Rewrite ${(x^{2} + 2)}^{18}$ as ${({(x^{2} + 2)}^{3})}^{6}$ .

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

Step 2

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

${(x^{2} + 2)}^{3}$

Step 3

Use the Binomial Theorem.

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

Step 4

Simplify each term.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

Multiply $2$ by $3$ .

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

Step 4.2

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

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Step 4.2.1

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

$x^{6} + 3 x^{2 \cdot 2} \cdot 2 + 3 x^{2} \cdot 2^{2} + 2^{3}$

Step 4.2.2

Multiply $2$ by $2$ .

$x^{6} + 3 x^{4} \cdot 2 + 3 x^{2} \cdot 2^{2} + 2^{3}$

Step 4.3

Multiply $2$ by $3$ .

$x^{6} + 6 x^{4} + 3 x^{2} \cdot 2^{2} + 2^{3}$

Step 4.4

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

$x^{6} + 6 x^{4} + 3 x^{2} \cdot 4 + 2^{3}$

Step 4.5

Multiply $4$ by $3$ .

$x^{6} + 6 x^{4} + 12 x^{2} + 2^{3}$

Step 4.6

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

$x^{6} + 6 x^{4} + 12 x^{2} + 8$