Solve by Factoring 2x^(3/2)=16

$2 x^{\frac{3}{2}} = 16$

Step 1

Subtract $16$ from both sides of the equation.

$2 x^{\frac{3}{2}} - 16 = 0$

Step 2

Factor $2$ out of $2 x^{\frac{3}{2}} - 16$ .

Factor $2$ out of $2 x^{\frac{3}{2}}$ .

$2 (x^{\frac{3}{2}}) - 16 = 0$

Factor $2$ out of $- 16$ .

$2 x^{\frac{3}{2}} + 2 \cdot - 8 = 0$

Factor $2$ out of $2 x^{\frac{3}{2}} + 2 \cdot - 8$ .

$2 (x^{\frac{3}{2}} - 8) = 0$

Step 3

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

$2 ({(x^{\frac{1}{2}})}^{3} - 8) = 0$

Step 4

Rewrite $8$ as $2^{3}$ .

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

Step 5

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

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

Step 6

Factor.

Simplify.

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Move $2$ to the left of $x^{\frac{1}{2}}$ .

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

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

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

Remove unnecessary parentheses.

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

Step 7

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

$x^{\frac{1}{2}} - 2 = 0$

$x + 2 x^{\frac{1}{2}} + 4 = 0$

Step 8

Set $x^{\frac{1}{2}} - 2$ equal to $0$ and solve for $x$ .

Set $x^{\frac{1}{2}} - 2$ equal to $0$ .

$x^{\frac{1}{2}} - 2 = 0$

Solve $x^{\frac{1}{2}} - 2 = 0$ for $x$ .

Add $2$ to both sides of the equation.

$x^{\frac{1}{2}} = 2$

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

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

Simplify the exponent.

Simplify the left side.

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

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

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

$x^{\frac{1}{2} \cdot 2} = 2^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = 2^{2}$

Rewrite the expression.

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

Simplify.

$x = 2^{2}$

Simplify the right side.

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

$x = 4$

Step 9

Set $x + 2 x^{\frac{1}{2}} + 4$ equal to $0$ and solve for $x$ .

Set $x + 2 x^{\frac{1}{2}} + 4$ equal to $0$ .

$x + 2 x^{\frac{1}{2}} + 4 = 0$

Solve $x + 2 x^{\frac{1}{2}} + 4 = 0$ for $x$ .

Find a common factor $x^{\frac{1}{2}}$ that is present in each term.

${(x^{\frac{1}{2}})}^{2} + 2 x^{\frac{1}{2}} + 4$

Substitute $u$ for $x^{\frac{1}{2}}$ .

${(u)}^{2} + 2 (u) + 4 = 0$

Solve for $u$ .

Multiply $2$ by $u$ .

$u^{2} + 2 u + 4 = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 1$ , $b = 2$ , and $c = 4$ into the quadratic formula and solve for $u$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (1 \cdot 4)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

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

$u = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 4$ .

Multiply $- 4$ by $1$ .

$u = \frac{- 2 \pm \sqrt{4 - 4 \cdot 4}}{2 \cdot 1}$

Multiply $- 4$ by $4$ .

$u = \frac{- 2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

Subtract $16$ from $4$ .

$u = \frac{- 2 \pm \sqrt{- 12}}{2 \cdot 1}$

Rewrite $- 12$ as $- 1 (12)$ .

$u = \frac{- 2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$u = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$u = \frac{- 2 \pm i \cdot \sqrt{12}}{2 \cdot 1}$

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

$u = \frac{- 2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 1}$

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

$u = \frac{- 2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$u = \frac{- 2 \pm i \cdot (2 \sqrt{3})}{2 \cdot 1}$

Move $2$ to the left of $i$ .

$u = \frac{- 2 \pm 2 i \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$u = \frac{- 2 \pm 2 i \sqrt{3}}{2}$

Simplify $\frac{- 2 \pm 2 i \sqrt{3}}{2}$ .

$u = - 1 \pm i \sqrt{3}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

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

$u = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 4$ .

Multiply $- 4$ by $1$ .

$u = \frac{- 2 \pm \sqrt{4 - 4 \cdot 4}}{2 \cdot 1}$

Multiply $- 4$ by $4$ .

$u = \frac{- 2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

Subtract $16$ from $4$ .

$u = \frac{- 2 \pm \sqrt{- 12}}{2 \cdot 1}$

Rewrite $- 12$ as $- 1 (12)$ .

$u = \frac{- 2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$u = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$u = \frac{- 2 \pm i \cdot \sqrt{12}}{2 \cdot 1}$

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

$u = \frac{- 2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 1}$

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

$u = \frac{- 2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$u = \frac{- 2 \pm i \cdot (2 \sqrt{3})}{2 \cdot 1}$

Move $2$ to the left of $i$ .

$u = \frac{- 2 \pm 2 i \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$u = \frac{- 2 \pm 2 i \sqrt{3}}{2}$

Simplify $\frac{- 2 \pm 2 i \sqrt{3}}{2}$ .

$u = - 1 \pm i \sqrt{3}$

Change the $\pm$ to $+$ .

$u = - 1 + i \sqrt{3}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

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

$u = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 4$ .

Multiply $- 4$ by $1$ .

$u = \frac{- 2 \pm \sqrt{4 - 4 \cdot 4}}{2 \cdot 1}$

Multiply $- 4$ by $4$ .

$u = \frac{- 2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

Subtract $16$ from $4$ .

$u = \frac{- 2 \pm \sqrt{- 12}}{2 \cdot 1}$

Rewrite $- 12$ as $- 1 (12)$ .

$u = \frac{- 2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$u = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$u = \frac{- 2 \pm i \cdot \sqrt{12}}{2 \cdot 1}$

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

$u = \frac{- 2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 1}$

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

$u = \frac{- 2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$u = \frac{- 2 \pm i \cdot (2 \sqrt{3})}{2 \cdot 1}$

Move $2$ to the left of $i$ .

$u = \frac{- 2 \pm 2 i \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$u = \frac{- 2 \pm 2 i \sqrt{3}}{2}$

Simplify $\frac{- 2 \pm 2 i \sqrt{3}}{2}$ .

$u = - 1 \pm i \sqrt{3}$

Change the $\pm$ to $-$ .

$u = - 1 - i \sqrt{3}$

The final answer is the combination of both solutions.

$u = - 1 + i \sqrt{3}, - 1 - i \sqrt{3}$

Substitute $x$ for $u$ .

$x^{\frac{1}{2}} = - 1 + i \sqrt{3}, - 1 - i \sqrt{3}$

Solve for $x^{\frac{1}{2}} = - 1 + i \sqrt{3}$ for $x$ .

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

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

Simplify the exponent.

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$x^{1} = {(- 1 + i \sqrt{3})}^{2}$

Simplify.

$x = {(- 1 + i \sqrt{3})}^{2}$

Simplify the right side.

Simplify ${(- 1 + i \sqrt{3})}^{2}$ .

Rewrite ${(- 1 + i \sqrt{3})}^{2}$ as $(- 1 + i \sqrt{3}) (- 1 + i \sqrt{3})$ .

$x = (- 1 + i \sqrt{3}) (- 1 + i \sqrt{3})$

Expand $(- 1 + i \sqrt{3}) (- 1 + i \sqrt{3})$ using the FOIL Method.

Apply the distributive property.

$x = - 1 (- 1 + i \sqrt{3}) + i \sqrt{3} (- 1 + i \sqrt{3})$

Apply the distributive property.

$x = - 1 \cdot - 1 - 1 (i \sqrt{3}) + i \sqrt{3} (- 1 + i \sqrt{3})$

Apply the distributive property.

$x = - 1 \cdot - 1 - 1 (i \sqrt{3}) + i \sqrt{3} \cdot - 1 + i \sqrt{3} (i \sqrt{3})$

Simplify and combine like terms.

Simplify each term.

Multiply $- 1$ by $- 1$ .

$x = 1 - 1 (i \sqrt{3}) + i \sqrt{3} \cdot - 1 + i \sqrt{3} (i \sqrt{3})$

Rewrite $- 1 (i \sqrt{3})$ as $- (i \sqrt{3})$ .

$x = 1 - (i \sqrt{3}) + i \sqrt{3} \cdot - 1 + i \sqrt{3} (i \sqrt{3})$

Move $- 1$ to the left of $i \sqrt{3}$ .

$x = 1 - i \sqrt{3} - 1 \cdot (i \sqrt{3}) + i \sqrt{3} (i \sqrt{3})$

Rewrite $- 1 (i \sqrt{3})$ as $- (i \sqrt{3})$ .

$x = 1 - i \sqrt{3} - (i \sqrt{3}) + i \sqrt{3} (i \sqrt{3})$

Multiply $i \sqrt{3} (i \sqrt{3})$ .

Raise $i$ to the power of $1$ .

$x = 1 - i \sqrt{3} - i \sqrt{3} + i^{1} i \sqrt{3} \sqrt{3}$

Raise $i$ to the power of $1$ .

$x = 1 - i \sqrt{3} - i \sqrt{3} + i^{1} i^{1} \sqrt{3} \sqrt{3}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = 1 - i \sqrt{3} - i \sqrt{3} + i^{1 + 1} \sqrt{3} \sqrt{3}$

Add $1$ and $1$ .

$x = 1 - i \sqrt{3} - i \sqrt{3} + i^{2} \sqrt{3} \sqrt{3}$

Raise $\sqrt{3}$ to the power of $1$ .

$x = 1 - i \sqrt{3} - i \sqrt{3} + i^{2} ({\sqrt{3}}^{1} \sqrt{3})$

Raise $\sqrt{3}$ to the power of $1$ .

$x = 1 - i \sqrt{3} - i \sqrt{3} + i^{2} ({\sqrt{3}}^{1} {\sqrt{3}}^{1})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = 1 - i \sqrt{3} - i \sqrt{3} + i^{2} {\sqrt{3}}^{1 + 1}$

Add $1$ and $1$ .

$x = 1 - i \sqrt{3} - i \sqrt{3} + i^{2} {\sqrt{3}}^{2}$

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

$x = 1 - i \sqrt{3} - i \sqrt{3} - 1 {\sqrt{3}}^{2}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = 1 - i \sqrt{3} - i \sqrt{3} - 1 {(3^{\frac{1}{2}})}^{2}$

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

$x = 1 - i \sqrt{3} - i \sqrt{3} - 1 \cdot 3^{\frac{1}{2} \cdot 2}$

Combine $\frac{1}{2}$ and $2$ .

$x = 1 - i \sqrt{3} - i \sqrt{3} - 1 \cdot 3^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = 1 - i \sqrt{3} - i \sqrt{3} - 1 \cdot 3^{\frac{2}{2}}$

Rewrite the expression.

$x = 1 - i \sqrt{3} - i \sqrt{3} - 1 \cdot 3^{1}$

Evaluate the exponent.

$x = 1 - i \sqrt{3} - i \sqrt{3} - 1 \cdot 3$

Multiply $- 1$ by $3$ .

$x = 1 - i \sqrt{3} - i \sqrt{3} - 3$

Subtract $3$ from $1$ .

$x = - i \sqrt{3} - i \sqrt{3} - 2$

Subtract $i \sqrt{3}$ from $- i \sqrt{3}$ .

$x = - 2 i \sqrt{3} - 2$

Reorder $- 2 i \sqrt{3}$ and $- 2$ .

$x = - 2 - 2 i \sqrt{3}$

Solve for $x^{\frac{1}{2}} = - 1 - i \sqrt{3}$ for $x$ .

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{2}})}^{2} = {(- 1 - i \sqrt{3})}^{2}$

Simplify the exponent.

Simplify the left side.

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

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

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

$x^{\frac{1}{2} \cdot 2} = {(- 1 - i \sqrt{3})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = {(- 1 - i \sqrt{3})}^{2}$

Rewrite the expression.

$x^{1} = {(- 1 - i \sqrt{3})}^{2}$

Simplify.

$x = {(- 1 - i \sqrt{3})}^{2}$

Simplify the right side.

Simplify ${(- 1 - i \sqrt{3})}^{2}$ .

Rewrite ${(- 1 - i \sqrt{3})}^{2}$ as $(- 1 - i \sqrt{3}) (- 1 - i \sqrt{3})$ .

$x = (- 1 - i \sqrt{3}) (- 1 - i \sqrt{3})$

Expand $(- 1 - i \sqrt{3}) (- 1 - i \sqrt{3})$ using the FOIL Method.

Apply the distributive property.

$x = - 1 (- 1 - i \sqrt{3}) - i \sqrt{3} (- 1 - i \sqrt{3})$

Apply the distributive property.

$x = - 1 \cdot - 1 - 1 (- i \sqrt{3}) - i \sqrt{3} (- 1 - i \sqrt{3})$

Apply the distributive property.

$x = - 1 \cdot - 1 - 1 (- i \sqrt{3}) - i \sqrt{3} \cdot - 1 - i \sqrt{3} (- i \sqrt{3})$

Simplify and combine like terms.

Simplify each term.

Multiply $- 1$ by $- 1$ .

$x = 1 - 1 (- i \sqrt{3}) - i \sqrt{3} \cdot - 1 - i \sqrt{3} (- i \sqrt{3})$

Multiply $- 1 (- i \sqrt{3})$ .

Multiply $- 1$ by $- 1$ .

$x = 1 + 1 (i \sqrt{3}) - i \sqrt{3} \cdot - 1 - i \sqrt{3} (- i \sqrt{3})$

Multiply $\sqrt{3}$ by $1$ .

$x = 1 + \sqrt{3} i - i \sqrt{3} \cdot - 1 - i \sqrt{3} (- i \sqrt{3})$

Multiply $- i \sqrt{3} \cdot - 1$ .

Multiply $- 1$ by $- 1$ .

$x = 1 + \sqrt{3} i + 1 i \sqrt{3} - i \sqrt{3} (- i \sqrt{3})$

Multiply $\sqrt{3}$ by $1$ .

$x = 1 + \sqrt{3} i + \sqrt{3} i - i \sqrt{3} (- i \sqrt{3})$

Multiply $- i \sqrt{3} (- i \sqrt{3})$ .

Multiply $- 1$ by $- 1$ .

$x = 1 + \sqrt{3} i + \sqrt{3} i + 1 i \sqrt{3} (i \sqrt{3})$

Multiply $\sqrt{3}$ by $1$ .

$x = 1 + \sqrt{3} i + \sqrt{3} i + \sqrt{3} i (i \sqrt{3})$

Raise $\sqrt{3}$ to the power of $1$ .

$x = 1 + \sqrt{3} i + \sqrt{3} i + {\sqrt{3}}^{1} \sqrt{3} i i$

Raise $\sqrt{3}$ to the power of $1$ .

$x = 1 + \sqrt{3} i + \sqrt{3} i + {\sqrt{3}}^{1} {\sqrt{3}}^{1} i i$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = 1 + \sqrt{3} i + \sqrt{3} i + {\sqrt{3}}^{1 + 1} i i$

Add $1$ and $1$ .

$x = 1 + \sqrt{3} i + \sqrt{3} i + {\sqrt{3}}^{2} i i$

Raise $i$ to the power of $1$ .

$x = 1 + \sqrt{3} i + \sqrt{3} i + {\sqrt{3}}^{2} (i^{1} i)$

Raise $i$ to the power of $1$ .

$x = 1 + \sqrt{3} i + \sqrt{3} i + {\sqrt{3}}^{2} (i^{1} i^{1})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = 1 + \sqrt{3} i + \sqrt{3} i + {\sqrt{3}}^{2} i^{1 + 1}$

Add $1$ and $1$ .

$x = 1 + \sqrt{3} i + \sqrt{3} i + {\sqrt{3}}^{2} i^{2}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

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

$x = 1 + \sqrt{3} i + \sqrt{3} i + 3^{\frac{1}{2} \cdot 2} i^{2}$

Combine $\frac{1}{2}$ and $2$ .

$x = 1 + \sqrt{3} i + \sqrt{3} i + 3^{\frac{2}{2}} i^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = 1 + \sqrt{3} i + \sqrt{3} i + 3^{\frac{2}{2}} i^{2}$

Rewrite the expression.

$x = 1 + \sqrt{3} i + \sqrt{3} i + 3^{1} i^{2}$

Evaluate the exponent.

$x = 1 + \sqrt{3} i + \sqrt{3} i + 3 i^{2}$

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

$x = 1 + \sqrt{3} i + \sqrt{3} i + 3 \cdot - 1$

Multiply $3$ by $- 1$ .

$x = 1 + \sqrt{3} i + \sqrt{3} i - 3$

Subtract $3$ from $1$ .

$x = - 2 + \sqrt{3} i + \sqrt{3} i$

Add $\sqrt{3} i$ and $\sqrt{3} i$ .

$x = - 2 + 2 \sqrt{3} i$

List all of the solutions.

$x = - 2 - 2 i \sqrt{3}, - 2 + 2 \sqrt{3} i$

Step 10

The final solution is all the values that make $2 (x^{\frac{1}{2}} - 2) (x + 2 x^{\frac{1}{2}} + 4) = 0$ true.

$x = 4, - 2 - 2 i \sqrt{3}, - 2 + 2 \sqrt{3} i$

Solve by Factoring 2x^(3/2)=16

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