Basic Math Examples

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

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

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.

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