Solve by Factoring fifth root of x^2+18x=-2

$\sqrt[5]{x^{2} + 18 x} = - 2$

Step 1

Add $2$ to both sides of the equation.

$\sqrt[5]{x^{2} + 18 x} + 2 = 0$

Step 2

Factor $x$ out of $x^{2} + 18 x$ .

Factor $x$ out of $x^{2}$ .

$\sqrt[5]{x \cdot x + 18 x} + 2 = 0$

Factor $x$ out of $18 x$ .

$\sqrt[5]{x \cdot x + x \cdot 18} + 2 = 0$

Factor $x$ out of $x \cdot x + x \cdot 18$ .

$\sqrt[5]{x (x + 18)} + 2 = 0$

Step 3

Subtract $2$ from both sides of the equation.

$\sqrt[5]{x (x + 18)} = - 2$

Step 4

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $5$ .

${\sqrt[5]{x (x + 18)}}^{5} = {(- 2)}^{5}$

Step 5

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x (x + 18)}$ as ${(x (x + 18))}^{\frac{1}{5}}$ .

${({(x (x + 18))}^{\frac{1}{5}})}^{5} = {(- 2)}^{5}$

Simplify the left side.

Simplify ${({(x (x + 18))}^{\frac{1}{5}})}^{5}$ .

Multiply the exponents in ${({(x (x + 18))}^{\frac{1}{5}})}^{5}$ .

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

${(x (x + 18))}^{\frac{1}{5} \cdot 5} = {(- 2)}^{5}$

Cancel the common factor of $5$ .

Cancel the common factor.

${(x (x + 18))}^{\frac{1}{5} \cdot 5} = {(- 2)}^{5}$

Rewrite the expression.

${(x (x + 18))}^{1} = {(- 2)}^{5}$

Apply the distributive property.

${(x \cdot x + x \cdot 18)}^{1} = {(- 2)}^{5}$

Simplify the expression.

Multiply $x$ by $x$ .

${(x^{2} + x \cdot 18)}^{1} = {(- 2)}^{5}$

Move $18$ to the left of $x$ .

$x^{2} + 18 x = {(- 2)}^{5}$

Simplify the right side.

Raise $- 2$ to the power of $5$ .

$x^{2} + 18 x = - 32$

Step 6

Solve for $x$ .

Add $32$ to both sides of the equation.

$x^{2} + 18 x + 32 = 0$

Factor $x^{2} + 18 x + 32$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $32$ and whose sum is $18$ .

$2, 16$

Write the factored form using these integers.

$(x + 2) (x + 16) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 2 = 0$

$x + 16 = 0$

Set $x + 2$ equal to $0$ and solve for $x$ .

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Set $x + 16$ equal to $0$ and solve for $x$ .

Set $x + 16$ equal to $0$ .

$x + 16 = 0$

Subtract $16$ from both sides of the equation.

$x = - 16$

The final solution is all the values that make $(x + 2) (x + 16) = 0$ true.

$x = - 2, - 16$

Solve by Factoring fifth root of x^2+18x=-2

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