Find All Complex Solutions (x+6)^(2/3)=4

${(x + 6)}^{\frac{2}{3}} = 4$

Step 1

Subtract $4$ from both sides of the equation.

${(x + 6)}^{\frac{2}{3}} - 4 = 0$

Step 2

Factor the left side of the equation.

Rewrite ${(x + 6)}^{\frac{2}{3}}$ as ${({(x + 6)}^{\frac{1}{3}})}^{2}$ .

${({(x + 6)}^{\frac{1}{3}})}^{2} - 4 = 0$

Rewrite $4$ as $2^{2}$ .

${({(x + 6)}^{\frac{1}{3}})}^{2} - 2^{2} = 0$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = {(x + 6)}^{\frac{1}{3}}$ and $b = 2$ .

$({(x + 6)}^{\frac{1}{3}} + 2) ({(x + 6)}^{\frac{1}{3}} - 2) = 0$

Step 3

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

${(x + 6)}^{\frac{1}{3}} + 2 = 0$

${(x + 6)}^{\frac{1}{3}} - 2 = 0$

Step 4

Set ${(x + 6)}^{\frac{1}{3}} + 2$ equal to $0$ and solve for $x$ .

Set ${(x + 6)}^{\frac{1}{3}} + 2$ equal to $0$ .

${(x + 6)}^{\frac{1}{3}} + 2 = 0$

Solve ${(x + 6)}^{\frac{1}{3}} + 2 = 0$ for $x$ .

Subtract $2$ from both sides of the equation.

${(x + 6)}^{\frac{1}{3}} = - 2$

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

${({(x + 6)}^{\frac{1}{3}})}^{3} = {(- 2)}^{3}$

Simplify the exponent.

Simplify the left side.

Simplify ${({(x + 6)}^{\frac{1}{3}})}^{3}$ .

Multiply the exponents in ${({(x + 6)}^{\frac{1}{3}})}^{3}$ .

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

${(x + 6)}^{\frac{1}{3} \cdot 3} = {(- 2)}^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(x + 6)}^{\frac{1}{3} \cdot 3} = {(- 2)}^{3}$

Rewrite the expression.

${(x + 6)}^{1} = {(- 2)}^{3}$

Simplify.

$x + 6 = {(- 2)}^{3}$

Simplify the right side.

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

$x + 6 = - 8$

Move all terms not containing $x$ to the right side of the equation.

Subtract $6$ from both sides of the equation.

$x = - 8 - 6$

Subtract $6$ from $- 8$ .

$x = - 14$

Step 5

Set ${(x + 6)}^{\frac{1}{3}} - 2$ equal to $0$ and solve for $x$ .

Set ${(x + 6)}^{\frac{1}{3}} - 2$ equal to $0$ .

${(x + 6)}^{\frac{1}{3}} - 2 = 0$

Solve ${(x + 6)}^{\frac{1}{3}} - 2 = 0$ for $x$ .

Add $2$ to both sides of the equation.

${(x + 6)}^{\frac{1}{3}} = 2$

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

${({(x + 6)}^{\frac{1}{3}})}^{3} = 2^{3}$

Simplify the exponent.

Simplify the left side.

Simplify ${({(x + 6)}^{\frac{1}{3}})}^{3}$ .

Multiply the exponents in ${({(x + 6)}^{\frac{1}{3}})}^{3}$ .

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

${(x + 6)}^{\frac{1}{3} \cdot 3} = 2^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(x + 6)}^{\frac{1}{3} \cdot 3} = 2^{3}$

Rewrite the expression.

${(x + 6)}^{1} = 2^{3}$

Simplify.

$x + 6 = 2^{3}$

Simplify the right side.

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

$x + 6 = 8$

Move all terms not containing $x$ to the right side of the equation.

Subtract $6$ from both sides of the equation.

$x = 8 - 6$

Subtract $6$ from $8$ .

$x = 2$

Step 6

The final solution is all the values that make $({(x + 6)}^{\frac{1}{3}} + 2) ({(x + 6)}^{\frac{1}{3}} - 2) = 0$ true.

$x = - 14, 2$

Find All Complex Solutions (x+6)^(2/3)=4

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