Algebra Examples

\sqrt[5]{x^{2} + 2 x} = - 1

$\sqrt[5]{x^{2} + 2 x} = - 1$

Step 1

Add

1

$1$ to both sides of the equation.

\sqrt[5]{x^{2} + 2 x} + 1 = 0

$\sqrt[5]{x^{2} + 2 x} + 1 = 0$

Step 2

Factor

x

$x$ out of

x^{2} + 2 x

$x^{2} + 2 x$ .

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Step 2.1

Factor

x

$x$ out of

x^{2}

$x^{2}$ .

\sqrt[5]{x \cdot x + 2 x} + 1 = 0

$\sqrt[5]{x \cdot x + 2 x} + 1 = 0$

Step 2.2

Factor

x

$x$ out of

2 x

$2 x$ .

\sqrt[5]{x \cdot x + x \cdot 2} + 1 = 0

$\sqrt[5]{x \cdot x + x \cdot 2} + 1 = 0$

Step 2.3

Factor

x

$x$ out of

x \cdot x + x \cdot 2

$x \cdot x + x \cdot 2$ .

\sqrt[5]{x (x + 2)} + 1 = 0

$\sqrt[5]{x (x + 2)} + 1 = 0$

\sqrt[5]{x (x + 2)} + 1 = 0

$\sqrt[5]{x (x + 2)} + 1 = 0$

Step 3

Subtract

1

$1$ from both sides of the equation.

\sqrt[5]{x (x + 2)} = - 1

$\sqrt[5]{x (x + 2)} = - 1$

Step 4

To remove the radical on the left side of the equation, raise both sides of the equation to the power of

5

$5$ .

{\sqrt[5]{x (x + 2)}}^{5} = {(- 1)}^{5}

${\sqrt[5]{x (x + 2)}}^{5} = {(- 1)}^{5}$

Step 5

Simplify each side of the equation.

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Step 5.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt[5]{x (x + 2)}

$\sqrt[5]{x (x + 2)}$ as

{(x (x + 2))}^{\frac{1}{5}}

${(x (x + 2))}^{\frac{1}{5}}$ .

{({(x (x + 2))}^{\frac{1}{5}})}^{5} = {(- 1)}^{5}

${({(x (x + 2))}^{\frac{1}{5}})}^{5} = {(- 1)}^{5}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify

{({(x (x + 2))}^{\frac{1}{5}})}^{5}

${({(x (x + 2))}^{\frac{1}{5}})}^{5}$ .

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Step 5.2.1.1

Multiply the exponents in

{({(x (x + 2))}^{\frac{1}{5}})}^{5}

${({(x (x + 2))}^{\frac{1}{5}})}^{5}$ .

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Step 5.2.1.1.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

{(x (x + 2))}^{\frac{1}{5} \cdot 5} = {(- 1)}^{5}

${(x (x + 2))}^{\frac{1}{5} \cdot 5} = {(- 1)}^{5}$

Step 5.2.1.1.2

Cancel the common factor of

5

$5$ .

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Step 5.2.1.1.2.1

Cancel the common factor.

${(x (x + 2))}^{\frac{1}{5} \cdot 5} = {(- 1)}^{5}$

Step 5.2.1.1.2.2

Rewrite the expression.

${(x (x + 2))}^{1} = {(- 1)}^{5}$

Step 5.2.1.2

Apply the distributive property.

${(x \cdot x + x \cdot 2)}^{1} = {(- 1)}^{5}$

Step 5.2.1.3

Simplify the expression.

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Step 5.2.1.3.1

Multiply

$x$ by

$x$ .

${(x^{2} + x \cdot 2)}^{1} = {(- 1)}^{5}$

Step 5.2.1.3.2

Move

$2$ to the left of

$x$ .

$x^{2} + 2 x = {(- 1)}^{5}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Raise

$- 1$ to the power of

$5$ .

$x^{2} + 2 x = - 1$

Step 6

Solve for

$x$ .

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Step 6.1

Add

$1$ to both sides of the equation.

$x^{2} + 2 x + 1 = 0$

Step 6.2

Factor using the perfect square rule.

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Step 6.2.1

Rewrite

$1$ as

$1^{2}$ .

$x^{2} + 2 x + 1^{2} = 0$

Step 6.2.2

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 6.2.3

Rewrite the polynomial.

$x^{2} + 2 \cdot x \cdot 1 + 1^{2} = 0$

Step 6.2.4

Factor using the perfect square trinomial rule

$a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where

$a = x$ and

$b = 1$ .

${(x + 1)}^{2} = 0$

Step 6.3

Set the

$x + 1$ equal to

$0$ .

$x + 1 = 0$

Step 6.4

Subtract

$1$ from both sides of the equation.

$x = - 1$