Basic Math Examples

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

Step 1

Subtract $Q^{2}$ from both sides of the equation.

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

Step 2

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

$Q^{\frac{1}{2}} - 1 {(Q^{\frac{1}{2}})}^{4}$

Step 3

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

$(u) - 1 {(u)}^{4} = 0$

Step 4

Solve for $u$ .

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Step 4.1

Rewrite $- 1 u^{4}$ as $- u^{4}$ .

$u - u^{4} = 0$

Step 4.2

Factor the left side of the equation.

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Step 4.2.1

Factor $u$ out of $u - u^{4}$ .

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Step 4.2.1.1

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

$u - u^{4} = 0$

Step 4.2.1.2

Factor $u$ out of $u^{1}$ .

$u \cdot 1 - u^{4} = 0$

Step 4.2.1.3

Factor $u$ out of $- u^{4}$ .

$u \cdot 1 + u (- u^{3}) = 0$

Step 4.2.1.4

Factor $u$ out of $u \cdot 1 + u (- u^{3})$ .

$u (1 - u^{3}) = 0$

Step 4.2.2

Rewrite $1$ as $1^{3}$ .

$u (1^{3} - u^{3}) = 0$

Step 4.2.3

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 $a = 1$ and $b = u$ .

$u ((1 - u) (1^{2} + 1 u + u^{2})) = 0$

Step 4.2.4

Factor.

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Step 4.2.4.1

Simplify.

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Step 4.2.4.1.1

One to any power is one.

$u ((1 - u) (1 + 1 u + u^{2})) = 0$

Step 4.2.4.1.2

Multiply $u$ by $1$ .

$u ((1 - u) (1 + u + u^{2})) = 0$

Step 4.2.4.2

Remove unnecessary parentheses.

$u (1 - u) (1 + u + u^{2}) = 0$

Step 4.3

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

$u = 0$

$1 - u = 0$

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

Step 4.4

Set $u$ equal to $0$ .

$u = 0$

Step 4.5

Set $1 - u$ equal to $0$ and solve for $u$ .

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Step 4.5.1

Set $1 - u$ equal to $0$ .

$1 - u = 0$

Step 4.5.2

Solve $1 - u = 0$ for $u$ .

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Step 4.5.2.1

Subtract $1$ from both sides of the equation.

$- u = - 1$

Step 4.5.2.2

Divide each term in $- u = - 1$ by $- 1$ and simplify.

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Step 4.5.2.2.1

Divide each term in $- u = - 1$ by $- 1$ .

$\frac{- u}{- 1} = \frac{- 1}{- 1}$

Step 4.5.2.2.2

Simplify the left side.

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Step 4.5.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{u}{1} = \frac{- 1}{- 1}$

Step 4.5.2.2.2.2

Divide $u$ by $1$ .

$u = \frac{- 1}{- 1}$

Step 4.5.2.2.3

Simplify the right side.

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Step 4.5.2.2.3.1

Divide $- 1$ by $- 1$ .

$u = 1$

Step 4.6

Set $1 + u + u^{2}$ equal to $0$ and solve for $u$ .

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Step 4.6.1

Set $1 + u + u^{2}$ equal to $0$ .

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

Step 4.6.2

Solve $1 + u + u^{2} = 0$ for $u$ .

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Step 4.6.2.1

Use the quadratic formula to find the solutions.

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

Step 4.6.2.2

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

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

Step 4.6.2.3

Simplify.

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Step 4.6.2.3.1

Simplify the numerator.

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Step 4.6.2.3.1.1

One to any power is one.

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

Step 4.6.2.3.1.2

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

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Step 4.6.2.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 4.6.2.3.1.2.2

Multiply $- 4$ by $1$ .

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

Step 4.6.2.3.1.3

Subtract $4$ from $1$ .

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

Step 4.6.2.3.1.4

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 4.6.2.3.1.5

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

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

Step 4.6.2.3.1.6

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

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

Step 4.6.2.3.2

Multiply $2$ by $1$ .

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

Step 4.6.2.4

The final answer is the combination of both solutions.

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

Step 4.7

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

$u = 0, 1, - \frac{1 - i \sqrt{3}}{2}, - \frac{1 + i \sqrt{3}}{2}$

Step 5

Substitute $Q$ for $u$ .

$Q^{\frac{1}{2}} = 0, 1, - \frac{1 - i \sqrt{3}}{2}, - \frac{1 + i \sqrt{3}}{2}$

Step 6

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

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Step 6.1

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

${(Q^{\frac{1}{2}})}^{2} = 0^{2}$

Step 6.2

Simplify the exponent.

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Step 6.2.1

Simplify the left side.

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Step 6.2.1.1

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

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Step 6.2.1.1.1

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

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Step 6.2.1.1.1.1

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

$Q^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 6.2.1.1.1.2

Cancel the common factor of $2$ .

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Step 6.2.1.1.1.2.1

Cancel the common factor.

$Q^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 6.2.1.1.1.2.2

Rewrite the expression.

$Q^{1} = 0^{2}$

Step 6.2.1.1.2

Simplify.

$Q = 0^{2}$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Raising $0$ to any positive power yields $0$ .

$Q = 0$

Step 7

Solve for $Q^{\frac{1}{2}} = 1$ for $Q$ .

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Step 7.1

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

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

Step 7.2

Simplify the exponent.

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Step 7.2.1

Simplify the left side.

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Step 7.2.1.1

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

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Step 7.2.1.1.1

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

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Step 7.2.1.1.1.1

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

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

Step 7.2.1.1.1.2

Cancel the common factor of $2$ .

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Step 7.2.1.1.1.2.1

Cancel the common factor.

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

Step 7.2.1.1.1.2.2

Rewrite the expression.

$Q^{1} = 1^{2}$

Step 7.2.1.1.2

Simplify.

$Q = 1^{2}$

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

One to any power is one.

$Q = 1$

Step 8

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

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Step 8.1

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

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

Step 8.2

Simplify the exponent.

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Step 8.2.1

Simplify the left side.

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Step 8.2.1.1

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

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Step 8.2.1.1.1

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

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Step 8.2.1.1.1.1

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

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

Step 8.2.1.1.1.2

Cancel the common factor of $2$ .

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Step 8.2.1.1.1.2.1

Cancel the common factor.

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

Step 8.2.1.1.1.2.2

Rewrite the expression.

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

Step 8.2.1.1.2

Simplify.

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

Step 8.2.2

Simplify the right side.

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Step 8.2.2.1

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

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Step 8.2.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 8.2.2.1.1.1

Apply the product rule to $- \frac{1 - i \sqrt{3}}{2}$ .

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

Step 8.2.2.1.1.2

Apply the product rule to $\frac{1 - i \sqrt{3}}{2}$ .

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

Step 8.2.2.1.2

Simplify the expression.

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Step 8.2.2.1.2.1

Raise $- 1$ to the power of $2$ .

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

Step 8.2.2.1.2.2

Multiply $\frac{{(1 - i \sqrt{3})}^{2}}{2^{2}}$ by $1$ .

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

Step 8.2.2.1.2.3

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

$Q = \frac{{(1 - i \sqrt{3})}^{2}}{4}$

Step 8.2.2.1.2.4

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

$Q = \frac{(1 - i \sqrt{3}) (1 - i \sqrt{3})}{4}$

Step 8.2.2.1.3

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

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Step 8.2.2.1.3.1

Apply the distributive property.

$Q = \frac{1 (1 - i \sqrt{3}) - i \sqrt{3} (1 - i \sqrt{3})}{4}$

Step 8.2.2.1.3.2

Apply the distributive property.

$Q = \frac{1 \cdot 1 + 1 (- i \sqrt{3}) - i \sqrt{3} (1 - i \sqrt{3})}{4}$

Step 8.2.2.1.3.3

Apply the distributive property.

$Q = \frac{1 \cdot 1 + 1 (- i \sqrt{3}) - i \sqrt{3} \cdot 1 - i \sqrt{3} (- i \sqrt{3})}{4}$

Step 8.2.2.1.4

Simplify and combine like terms.

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Step 8.2.2.1.4.1

Simplify each term.

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Step 8.2.2.1.4.1.1

Multiply $1$ by $1$ .

$Q = \frac{1 + 1 (- i \sqrt{3}) - i \sqrt{3} \cdot 1 - i \sqrt{3} (- i \sqrt{3})}{4}$

Step 8.2.2.1.4.1.2

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

$Q = \frac{1 - i \sqrt{3} - i \sqrt{3} \cdot 1 - i \sqrt{3} (- i \sqrt{3})}{4}$

Step 8.2.2.1.4.1.3

Multiply $- 1$ by $1$ .

$Q = \frac{1 - i \sqrt{3} - i \sqrt{3} - i \sqrt{3} (- i \sqrt{3})}{4}$

Step 8.2.2.1.4.1.4

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

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Step 8.2.2.1.4.1.4.1

Multiply $- 1$ by $- 1$ .

$Q = \frac{1 - i \sqrt{3} - i \sqrt{3} + 1 i \sqrt{3} (i \sqrt{3})}{4}$

Step 8.2.2.1.4.1.4.2

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

$Q = \frac{1 - i \sqrt{3} - i \sqrt{3} + \sqrt{3} i (i \sqrt{3})}{4}$

Step 8.2.2.1.4.1.4.3

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

$Q = \frac{1 - i \sqrt{3} - i \sqrt{3} + {\sqrt{3}}^{1} \sqrt{3} i i}{4}$

Step 8.2.2.1.4.1.4.4

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

$Q = \frac{1 - i \sqrt{3} - i \sqrt{3} + {\sqrt{3}}^{1} {\sqrt{3}}^{1} i i}{4}$

Step 8.2.2.1.4.1.4.5

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

$Q = \frac{1 - i \sqrt{3} - i \sqrt{3} + {\sqrt{3}}^{1 + 1} i i}{4}$

Step 8.2.2.1.4.1.4.6

Add $1$ and $1$ .

$Q = \frac{1 - i \sqrt{3} - i \sqrt{3} + {\sqrt{3}}^{2} i i}{4}$

Step 8.2.2.1.4.1.4.7

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

$Q = \frac{1 - i \sqrt{3} - i \sqrt{3} + {\sqrt{3}}^{2} (i^{1} i)}{4}$

Step 8.2.2.1.4.1.4.8

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

$Q = \frac{1 - i \sqrt{3} - i \sqrt{3} + {\sqrt{3}}^{2} (i^{1} i^{1})}{4}$

Step 8.2.2.1.4.1.4.9

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

$Q = \frac{1 - i \sqrt{3} - i \sqrt{3} + {\sqrt{3}}^{2} i^{1 + 1}}{4}$

Step 8.2.2.1.4.1.4.10

Add $1$ and $1$ .

$Q = \frac{1 - i \sqrt{3} - i \sqrt{3} + {\sqrt{3}}^{2} i^{2}}{4}$

Step 8.2.2.1.4.1.5

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

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Step 8.2.2.1.4.1.5.1

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

$Q = \frac{1 - i \sqrt{3} - i \sqrt{3} + {(3^{\frac{1}{2}})}^{2} i^{2}}{4}$

Step 8.2.2.1.4.1.5.2

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

$Q = \frac{1 - i \sqrt{3} - i \sqrt{3} + 3^{\frac{1}{2} \cdot 2} i^{2}}{4}$

Step 8.2.2.1.4.1.5.3

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

$Q = \frac{1 - i \sqrt{3} - i \sqrt{3} + 3^{\frac{2}{2}} i^{2}}{4}$

Step 8.2.2.1.4.1.5.4

Cancel the common factor of $2$ .

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Step 8.2.2.1.4.1.5.4.1

Cancel the common factor.

$Q = \frac{1 - i \sqrt{3} - i \sqrt{3} + 3^{\frac{2}{2}} i^{2}}{4}$

Step 8.2.2.1.4.1.5.4.2

Rewrite the expression.

$Q = \frac{1 - i \sqrt{3} - i \sqrt{3} + 3^{1} i^{2}}{4}$

Step 8.2.2.1.4.1.5.5

Evaluate the exponent.

$Q = \frac{1 - i \sqrt{3} - i \sqrt{3} + 3 i^{2}}{4}$

Step 8.2.2.1.4.1.6

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

$Q = \frac{1 - i \sqrt{3} - i \sqrt{3} + 3 \cdot - 1}{4}$

Step 8.2.2.1.4.1.7

Multiply $3$ by $- 1$ .

$Q = \frac{1 - i \sqrt{3} - i \sqrt{3} - 3}{4}$

Step 8.2.2.1.4.2

Subtract $3$ from $1$ .

$Q = \frac{- i \sqrt{3} - i \sqrt{3} - 2}{4}$

Step 8.2.2.1.4.3

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

$Q = \frac{- 2 i \sqrt{3} - 2}{4}$

Step 8.2.2.1.5

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

$Q = \frac{- 2 - 2 i \sqrt{3}}{4}$

Step 8.2.2.1.6

Cancel the common factor of $- 2 - 2 i \sqrt{3}$ and $4$ .

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Step 8.2.2.1.6.1

Factor $2$ out of $- 2$ .

$Q = \frac{2 (- 1) - 2 i \sqrt{3}}{4}$

Step 8.2.2.1.6.2

Factor $2$ out of $- 2 i \sqrt{3}$ .

$Q = \frac{2 (- 1) + 2 (- i \sqrt{3})}{4}$

Step 8.2.2.1.6.3

Factor $2$ out of $2 (- 1) + 2 (- i \sqrt{3})$ .

$Q = \frac{2 (- 1 - i \sqrt{3})}{4}$

Step 8.2.2.1.6.4

Cancel the common factors.

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Step 8.2.2.1.6.4.1

Factor $2$ out of $4$ .

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

Step 8.2.2.1.6.4.2

Cancel the common factor.

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

Step 8.2.2.1.6.4.3

Rewrite the expression.

$Q = \frac{- 1 - i \sqrt{3}}{2}$

Step 8.2.2.1.7

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

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

Step 8.2.2.1.8

Factor $- 1$ out of $- i \sqrt{3}$ .

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

Step 8.2.2.1.9

Factor $- 1$ out of $- 1 (1) - (i \sqrt{3})$ .

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

Step 8.2.2.1.10

Move the negative in front of the fraction.

$Q = - \frac{1 + i \sqrt{3}}{2}$

Step 9

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

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Step 9.1

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

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

Step 9.2

Simplify the exponent.

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Step 9.2.1

Simplify the left side.

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Step 9.2.1.1

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

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Step 9.2.1.1.1

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

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Step 9.2.1.1.1.1

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

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

Step 9.2.1.1.1.2

Cancel the common factor of $2$ .

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Step 9.2.1.1.1.2.1

Cancel the common factor.

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

Step 9.2.1.1.1.2.2

Rewrite the expression.

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

Step 9.2.1.1.2

Simplify.

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

Step 9.2.2

Simplify the right side.

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Step 9.2.2.1

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

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Step 9.2.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 9.2.2.1.1.1

Apply the product rule to $- \frac{1 + i \sqrt{3}}{2}$ .

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

Step 9.2.2.1.1.2

Apply the product rule to $\frac{1 + i \sqrt{3}}{2}$ .

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

Step 9.2.2.1.2

Simplify the expression.

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Step 9.2.2.1.2.1

Raise $- 1$ to the power of $2$ .

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

Step 9.2.2.1.2.2

Multiply $\frac{{(1 + i \sqrt{3})}^{2}}{2^{2}}$ by $1$ .

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

Step 9.2.2.1.2.3

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

$Q = \frac{{(1 + i \sqrt{3})}^{2}}{4}$

Step 9.2.2.1.2.4

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

$Q = \frac{(1 + i \sqrt{3}) (1 + i \sqrt{3})}{4}$

Step 9.2.2.1.3

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

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Step 9.2.2.1.3.1

Apply the distributive property.

$Q = \frac{1 (1 + i \sqrt{3}) + i \sqrt{3} (1 + i \sqrt{3})}{4}$

Step 9.2.2.1.3.2

Apply the distributive property.

$Q = \frac{1 \cdot 1 + 1 (i \sqrt{3}) + i \sqrt{3} (1 + i \sqrt{3})}{4}$

Step 9.2.2.1.3.3

Apply the distributive property.

$Q = \frac{1 \cdot 1 + 1 (i \sqrt{3}) + i \sqrt{3} \cdot 1 + i \sqrt{3} (i \sqrt{3})}{4}$

Step 9.2.2.1.4

Simplify and combine like terms.

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Step 9.2.2.1.4.1

Simplify each term.

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Step 9.2.2.1.4.1.1

Multiply $1$ by $1$ .

$Q = \frac{1 + 1 (i \sqrt{3}) + i \sqrt{3} \cdot 1 + i \sqrt{3} (i \sqrt{3})}{4}$

Step 9.2.2.1.4.1.2

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

$Q = \frac{1 + i \sqrt{3} + i \sqrt{3} \cdot 1 + i \sqrt{3} (i \sqrt{3})}{4}$

Step 9.2.2.1.4.1.3

Multiply $i$ by $1$ .

$Q = \frac{1 + i \sqrt{3} + i \sqrt{3} + i \sqrt{3} (i \sqrt{3})}{4}$

Step 9.2.2.1.4.1.4

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

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Step 9.2.2.1.4.1.4.1

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

$Q = \frac{1 + i \sqrt{3} + i \sqrt{3} + i^{1} i \sqrt{3} \sqrt{3}}{4}$

Step 9.2.2.1.4.1.4.2

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

$Q = \frac{1 + i \sqrt{3} + i \sqrt{3} + i^{1} i^{1} \sqrt{3} \sqrt{3}}{4}$

Step 9.2.2.1.4.1.4.3

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

$Q = \frac{1 + i \sqrt{3} + i \sqrt{3} + i^{1 + 1} \sqrt{3} \sqrt{3}}{4}$

Step 9.2.2.1.4.1.4.4

Add $1$ and $1$ .

$Q = \frac{1 + i \sqrt{3} + i \sqrt{3} + i^{2} \sqrt{3} \sqrt{3}}{4}$

Step 9.2.2.1.4.1.4.5

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

$Q = \frac{1 + i \sqrt{3} + i \sqrt{3} + i^{2} ({\sqrt{3}}^{1} \sqrt{3})}{4}$

Step 9.2.2.1.4.1.4.6

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

$Q = \frac{1 + i \sqrt{3} + i \sqrt{3} + i^{2} ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}{4}$

Step 9.2.2.1.4.1.4.7

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

$Q = \frac{1 + i \sqrt{3} + i \sqrt{3} + i^{2} {\sqrt{3}}^{1 + 1}}{4}$

Step 9.2.2.1.4.1.4.8

Add $1$ and $1$ .

$Q = \frac{1 + i \sqrt{3} + i \sqrt{3} + i^{2} {\sqrt{3}}^{2}}{4}$

Step 9.2.2.1.4.1.5

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

$Q = \frac{1 + i \sqrt{3} + i \sqrt{3} - 1 {\sqrt{3}}^{2}}{4}$

Step 9.2.2.1.4.1.6

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

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Step 9.2.2.1.4.1.6.1

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

$Q = \frac{1 + i \sqrt{3} + i \sqrt{3} - 1 {(3^{\frac{1}{2}})}^{2}}{4}$

Step 9.2.2.1.4.1.6.2

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

$Q = \frac{1 + i \sqrt{3} + i \sqrt{3} - 1 \cdot 3^{\frac{1}{2} \cdot 2}}{4}$

Step 9.2.2.1.4.1.6.3

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

$Q = \frac{1 + i \sqrt{3} + i \sqrt{3} - 1 \cdot 3^{\frac{2}{2}}}{4}$

Step 9.2.2.1.4.1.6.4

Cancel the common factor of $2$ .

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Step 9.2.2.1.4.1.6.4.1

Cancel the common factor.

$Q = \frac{1 + i \sqrt{3} + i \sqrt{3} - 1 \cdot 3^{\frac{2}{2}}}{4}$

Step 9.2.2.1.4.1.6.4.2

Rewrite the expression.

$Q = \frac{1 + i \sqrt{3} + i \sqrt{3} - 1 \cdot 3^{1}}{4}$

Step 9.2.2.1.4.1.6.5

Evaluate the exponent.

$Q = \frac{1 + i \sqrt{3} + i \sqrt{3} - 1 \cdot 3}{4}$

Step 9.2.2.1.4.1.7

Multiply $- 1$ by $3$ .

$Q = \frac{1 + i \sqrt{3} + i \sqrt{3} - 3}{4}$

Step 9.2.2.1.4.2

Subtract $3$ from $1$ .

$Q = \frac{i \sqrt{3} + i \sqrt{3} - 2}{4}$

Step 9.2.2.1.4.3

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

$Q = \frac{2 i \sqrt{3} - 2}{4}$

Step 9.2.2.1.5

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

$Q = \frac{- 2 + 2 i \sqrt{3}}{4}$

Step 9.2.2.1.6

Cancel the common factor of $- 2 + 2 i \sqrt{3}$ and $4$ .

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Step 9.2.2.1.6.1

Factor $2$ out of $- 2$ .

$Q = \frac{2 \cdot - 1 + 2 i \sqrt{3}}{4}$

Step 9.2.2.1.6.2

Factor $2$ out of $2 i \sqrt{3}$ .

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

Step 9.2.2.1.6.3

Factor $2$ out of $2 \cdot - 1 + 2 (i \sqrt{3})$ .

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

Step 9.2.2.1.6.4

Cancel the common factors.

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Step 9.2.2.1.6.4.1

Factor $2$ out of $4$ .

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

Step 9.2.2.1.6.4.2

Cancel the common factor.

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

Step 9.2.2.1.6.4.3

Rewrite the expression.

$Q = \frac{- 1 + i \sqrt{3}}{2}$

Step 9.2.2.1.7

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

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

Step 9.2.2.1.8

Factor $- 1$ out of $i \sqrt{3}$ .

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

Step 9.2.2.1.9

Factor $- 1$ out of $- 1 (1) - (- i \sqrt{3})$ .

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

Step 9.2.2.1.10

Move the negative in front of the fraction.

$Q = - \frac{1 - i \sqrt{3}}{2}$

Step 10

List all of the solutions.

$Q = 0, 1, - \frac{1 + i \sqrt{3}}{2}, - \frac{1 - i \sqrt{3}}{2}$