Algebra Examples

$\sqrt[4]{{(x^{2} + 2 x + 1)}^{2}}$

Step 1

Factor using the perfect square rule.

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Step 1.1

Rewrite $1$ as $1^{2}$ .

$\sqrt[4]{{(x^{2} + 2 x + 1^{2})}^{2}}$

Step 1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 1.3

Rewrite the polynomial.

$\sqrt[4]{{(x^{2} + 2 \cdot x \cdot 1 + 1^{2})}^{2}}$

Step 1.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 1$ .

$\sqrt[4]{{({(x + 1)}^{2})}^{2}}$

Step 2

Multiply the exponents in ${({(x + 1)}^{2})}^{2}$ .

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Step 2.1

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

$\sqrt[4]{{(x + 1)}^{2 \cdot 2}}$

Step 2.2

Multiply $2$ by $2$ .

$\sqrt[4]{{(x + 1)}^{4}}$

Step 3

Use the Binomial Theorem.

$\sqrt[4]{x^{4} + 4 x^{3} \cdot 1 + 6 x^{2} \cdot 1^{2} + 4 x \cdot 1^{3} + 1^{4}}$

Step 4

Simplify each term.

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Step 4.1

Multiply $4$ by $1$ .

$\sqrt[4]{x^{4} + 4 x^{3} + 6 x^{2} \cdot 1^{2} + 4 x \cdot 1^{3} + 1^{4}}$

Step 4.2

One to any power is one.

$\sqrt[4]{x^{4} + 4 x^{3} + 6 x^{2} \cdot 1 + 4 x \cdot 1^{3} + 1^{4}}$

Step 4.3

Multiply $6$ by $1$ .

$\sqrt[4]{x^{4} + 4 x^{3} + 6 x^{2} + 4 x \cdot 1^{3} + 1^{4}}$

Step 4.4

One to any power is one.

$\sqrt[4]{x^{4} + 4 x^{3} + 6 x^{2} + 4 x \cdot 1 + 1^{4}}$

Step 4.5

Multiply $4$ by $1$ .

$\sqrt[4]{x^{4} + 4 x^{3} + 6 x^{2} + 4 x + 1^{4}}$

Step 4.6

One to any power is one.

$\sqrt[4]{x^{4} + 4 x^{3} + 6 x^{2} + 4 x + 1}$

Step 5

Factor $x^{4} + 4 x^{3} + 6 x^{2} + 4 x + 1$ using the binomial theorem.

$\sqrt[4]{{(x + 1)}^{4}}$

Step 6

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

$x + 1$