Finite Math Examples

$m^{\frac{2}{3}} + 3 m^{\frac{1}{3}} + 8 = 0$

Step 1

Find a common factor $m^{\frac{1}{3}}$ that is present in each term.

${(m^{\frac{1}{3}})}^{2} + 3 m^{\frac{1}{3}} + 8$

Step 2

Substitute $u$ for $m^{\frac{1}{3}}$ .

${(u)}^{2} + 3 (u) + 8 = 0$

Step 3

Solve for $u$ .

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Step 3.1

Multiply $3$ by $u$ .

$u^{2} + 3 u + 8 = 0$

Step 3.2

Use the quadratic formula to find the solutions.

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

Step 3.3

Substitute the values $a = 1$ , $b = 3$ , and $c = 8$ into the quadratic formula and solve for $u$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (1 \cdot 8)}}{2 \cdot 1}$

Step 3.4

Simplify.

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Step 3.4.1

Simplify the numerator.

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Step 3.4.1.1

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

$u = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 8}}{2 \cdot 1}$

Step 3.4.1.2

Multiply $- 4 \cdot 1 \cdot 8$ .

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Step 3.4.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{- 3 \pm \sqrt{9 - 4 \cdot 8}}{2 \cdot 1}$

Step 3.4.1.2.2

Multiply $- 4$ by $8$ .

$u = \frac{- 3 \pm \sqrt{9 - 32}}{2 \cdot 1}$

Step 3.4.1.3

Subtract $32$ from $9$ .

$u = \frac{- 3 \pm \sqrt{- 23}}{2 \cdot 1}$

Step 3.4.1.4

Rewrite $- 23$ as $- 1 (23)$ .

$u = \frac{- 3 \pm \sqrt{- 1 \cdot 23}}{2 \cdot 1}$

Step 3.4.1.5

Rewrite $\sqrt{- 1 (23)}$ as $\sqrt{- 1} \cdot \sqrt{23}$ .

$u = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{23}}{2 \cdot 1}$

Step 3.4.1.6

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

$u = \frac{- 3 \pm i \sqrt{23}}{2 \cdot 1}$

Step 3.4.2

Multiply $2$ by $1$ .

$u = \frac{- 3 \pm i \sqrt{23}}{2}$

Step 3.5

The final answer is the combination of both solutions.

$u = - \frac{3 - i \sqrt{23}}{2}, - \frac{3 + i \sqrt{23}}{2}$

Step 4

Substitute $m$ for $u$ .

$m^{\frac{1}{3}} = - \frac{3 - i \sqrt{23}}{2}, - \frac{3 + i \sqrt{23}}{2}$

Step 5

Solve for $m^{\frac{1}{3}} = - \frac{3 - i \sqrt{23}}{2}$ for $m$ .

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Step 5.1

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

${(m^{\frac{1}{3}})}^{3} = {(- \frac{3 - i \sqrt{23}}{2})}^{3}$

Step 5.2

Simplify the exponent.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Simplify ${(m^{\frac{1}{3}})}^{3}$ .

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Step 5.2.1.1.1

Multiply the exponents in ${(m^{\frac{1}{3}})}^{3}$ .

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Step 5.2.1.1.1.1

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

$m^{\frac{1}{3} \cdot 3} = {(- \frac{3 - i \sqrt{23}}{2})}^{3}$

Step 5.2.1.1.1.2

Cancel the common factor of $3$ .

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Step 5.2.1.1.1.2.1

Cancel the common factor.

$m^{\frac{1}{3} \cdot 3} = {(- \frac{3 - i \sqrt{23}}{2})}^{3}$

Step 5.2.1.1.1.2.2

Rewrite the expression.

$m^{1} = {(- \frac{3 - i \sqrt{23}}{2})}^{3}$

Step 5.2.1.1.2

Simplify.

$m = {(- \frac{3 - i \sqrt{23}}{2})}^{3}$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify ${(- \frac{3 - i \sqrt{23}}{2})}^{3}$ .

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Step 5.2.2.1.1

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

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Step 5.2.2.1.1.1

Apply the product rule to $- \frac{3 - i \sqrt{23}}{2}$ .

$m = {(- 1)}^{3} {(\frac{3 - i \sqrt{23}}{2})}^{3}$

Step 5.2.2.1.1.2

Apply the product rule to $\frac{3 - i \sqrt{23}}{2}$ .

$m = {(- 1)}^{3} \frac{{(3 - i \sqrt{23})}^{3}}{2^{3}}$

Step 5.2.2.1.2

Evaluate the exponents.

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Step 5.2.2.1.2.1

Raise $- 1$ to the power of $3$ .

$m = - \frac{{(3 - i \sqrt{23})}^{3}}{2^{3}}$

Step 5.2.2.1.2.2

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

$m = - \frac{{(3 - i \sqrt{23})}^{3}}{8}$

Step 5.2.2.1.3

Use the Binomial Theorem.

$m = - \frac{3^{3} + 3 \cdot 3^{2} (- i \sqrt{23}) + 3 \cdot 3 {(- i \sqrt{23})}^{2} + {(- i \sqrt{23})}^{3}}{8}$

Step 5.2.2.1.4

Simplify terms.

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Step 5.2.2.1.4.1

Simplify each term.

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Step 5.2.2.1.4.1.1

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

$m = - \frac{27 + 3 \cdot 3^{2} (- i \sqrt{23}) + 3 \cdot 3 {(- i \sqrt{23})}^{2} + {(- i \sqrt{23})}^{3}}{8}$

Step 5.2.2.1.4.1.2

Multiply $3$ by $3^{2}$ by adding the exponents.

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Step 5.2.2.1.4.1.2.1

Multiply $3$ by $3^{2}$ .

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Step 5.2.2.1.4.1.2.1.1

Raise $3$ to the power of $1$ .

$m = - \frac{27 + 3^{1} \cdot 3^{2} (- i \sqrt{23}) + 3 \cdot 3 {(- i \sqrt{23})}^{2} + {(- i \sqrt{23})}^{3}}{8}$

Step 5.2.2.1.4.1.2.1.2

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

$m = - \frac{27 + 3^{1 + 2} (- i \sqrt{23}) + 3 \cdot 3 {(- i \sqrt{23})}^{2} + {(- i \sqrt{23})}^{3}}{8}$

Step 5.2.2.1.4.1.2.2

Add $1$ and $2$ .

$m = - \frac{27 + 3^{3} (- i \sqrt{23}) + 3 \cdot 3 {(- i \sqrt{23})}^{2} + {(- i \sqrt{23})}^{3}}{8}$

Step 5.2.2.1.4.1.3

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

$m = - \frac{27 + 27 (- i \sqrt{23}) + 3 \cdot 3 {(- i \sqrt{23})}^{2} + {(- i \sqrt{23})}^{3}}{8}$

Step 5.2.2.1.4.1.4

Multiply $- 1$ by $27$ .

$m = - \frac{27 - 27 (i \sqrt{23}) + 3 \cdot 3 {(- i \sqrt{23})}^{2} + {(- i \sqrt{23})}^{3}}{8}$

Step 5.2.2.1.4.1.5

Multiply $3$ by $3$ .

$m = - \frac{27 - 27 i \sqrt{23} + 9 {(- i \sqrt{23})}^{2} + {(- i \sqrt{23})}^{3}}{8}$

Step 5.2.2.1.4.1.6

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

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Step 5.2.2.1.4.1.6.1

Apply the product rule to $- i \sqrt{23}$ .

$m = - \frac{27 - 27 i \sqrt{23} + 9 ({(- i)}^{2} {\sqrt{23}}^{2}) + {(- i \sqrt{23})}^{3}}{8}$

Step 5.2.2.1.4.1.6.2

Apply the product rule to $- i$ .

$m = - \frac{27 - 27 i \sqrt{23} + 9 ({(- 1)}^{2} i^{2} {\sqrt{23}}^{2}) + {(- i \sqrt{23})}^{3}}{8}$

Step 5.2.2.1.4.1.7

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

$m = - \frac{27 - 27 i \sqrt{23} + 9 (1 i^{2} {\sqrt{23}}^{2}) + {(- i \sqrt{23})}^{3}}{8}$

Step 5.2.2.1.4.1.8

Multiply $i^{2}$ by $1$ .

$m = - \frac{27 - 27 i \sqrt{23} + 9 (i^{2} {\sqrt{23}}^{2}) + {(- i \sqrt{23})}^{3}}{8}$

Step 5.2.2.1.4.1.9

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

$m = - \frac{27 - 27 i \sqrt{23} + 9 (- 1 {\sqrt{23}}^{2}) + {(- i \sqrt{23})}^{3}}{8}$

Step 5.2.2.1.4.1.10

Rewrite ${\sqrt{23}}^{2}$ as $23$ .

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Step 5.2.2.1.4.1.10.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{23}$ as $23^{\frac{1}{2}}$ .

$m = - \frac{27 - 27 i \sqrt{23} + 9 (- 1 {(23^{\frac{1}{2}})}^{2}) + {(- i \sqrt{23})}^{3}}{8}$

Step 5.2.2.1.4.1.10.2

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

$m = - \frac{27 - 27 i \sqrt{23} + 9 (- 1 \cdot 23^{\frac{1}{2} \cdot 2}) + {(- i \sqrt{23})}^{3}}{8}$

Step 5.2.2.1.4.1.10.3

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

$m = - \frac{27 - 27 i \sqrt{23} + 9 (- 1 \cdot 23^{\frac{2}{2}}) + {(- i \sqrt{23})}^{3}}{8}$

Step 5.2.2.1.4.1.10.4

Cancel the common factor of $2$ .

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Step 5.2.2.1.4.1.10.4.1

Cancel the common factor.

$m = - \frac{27 - 27 i \sqrt{23} + 9 (- 1 \cdot 23^{\frac{2}{2}}) + {(- i \sqrt{23})}^{3}}{8}$

Step 5.2.2.1.4.1.10.4.2

Rewrite the expression.

$m = - \frac{27 - 27 i \sqrt{23} + 9 (- 1 \cdot 23^{1}) + {(- i \sqrt{23})}^{3}}{8}$

Step 5.2.2.1.4.1.10.5

Evaluate the exponent.

$m = - \frac{27 - 27 i \sqrt{23} + 9 (- 1 \cdot 23) + {(- i \sqrt{23})}^{3}}{8}$

Step 5.2.2.1.4.1.11

Multiply $9 (- 1 \cdot 23)$ .

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Step 5.2.2.1.4.1.11.1

Multiply $- 1$ by $23$ .

$m = - \frac{27 - 27 i \sqrt{23} + 9 \cdot - 23 + {(- i \sqrt{23})}^{3}}{8}$

Step 5.2.2.1.4.1.11.2

Multiply $9$ by $- 23$ .

$m = - \frac{27 - 27 i \sqrt{23} - 207 + {(- i \sqrt{23})}^{3}}{8}$

Step 5.2.2.1.4.1.12

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

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Step 5.2.2.1.4.1.12.1

Apply the product rule to $- i \sqrt{23}$ .

$m = - \frac{27 - 27 i \sqrt{23} - 207 + {(- i)}^{3} {\sqrt{23}}^{3}}{8}$

Step 5.2.2.1.4.1.12.2

Apply the product rule to $- i$ .

$m = - \frac{27 - 27 i \sqrt{23} - 207 + {(- 1)}^{3} i^{3} {\sqrt{23}}^{3}}{8}$

Step 5.2.2.1.4.1.13

Raise $- 1$ to the power of $3$ .

$m = - \frac{27 - 27 i \sqrt{23} - 207 - i^{3} {\sqrt{23}}^{3}}{8}$

Step 5.2.2.1.4.1.14

Factor out $i^{2}$ .

$m = - \frac{27 - 27 i \sqrt{23} - 207 - (i^{2} \cdot i) {\sqrt{23}}^{3}}{8}$

Step 5.2.2.1.4.1.15

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

$m = - \frac{27 - 27 i \sqrt{23} - 207 - (- 1 \cdot i) {\sqrt{23}}^{3}}{8}$

Step 5.2.2.1.4.1.16

Rewrite $- 1 i$ as $- i$ .

$m = - \frac{27 - 27 i \sqrt{23} - 207 - - i {\sqrt{23}}^{3}}{8}$

Step 5.2.2.1.4.1.17

Multiply $- 1$ by $- 1$ .

$m = - \frac{27 - 27 i \sqrt{23} - 207 + 1 i {\sqrt{23}}^{3}}{8}$

Step 5.2.2.1.4.1.18

Multiply $i$ by $1$ .

$m = - \frac{27 - 27 i \sqrt{23} - 207 + i {\sqrt{23}}^{3}}{8}$

Step 5.2.2.1.4.1.19

Rewrite ${\sqrt{23}}^{3}$ as $\sqrt{23^{3}}$ .

$m = - \frac{27 - 27 i \sqrt{23} - 207 + i \sqrt{23^{3}}}{8}$

Step 5.2.2.1.4.1.20

Raise $23$ to the power of $3$ .

$m = - \frac{27 - 27 i \sqrt{23} - 207 + i \sqrt{12167}}{8}$

Step 5.2.2.1.4.1.21

Rewrite $12167$ as $23^{2} \cdot 23$ .

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Step 5.2.2.1.4.1.21.1

Factor $529$ out of $12167$ .

$m = - \frac{27 - 27 i \sqrt{23} - 207 + i \sqrt{529 (23)}}{8}$

Step 5.2.2.1.4.1.21.2

Rewrite $529$ as $23^{2}$ .

$m = - \frac{27 - 27 i \sqrt{23} - 207 + i \sqrt{23^{2} \cdot 23}}{8}$

Step 5.2.2.1.4.1.22

Pull terms out from under the radical.

$m = - \frac{27 - 27 i \sqrt{23} - 207 + i (23 \sqrt{23})}{8}$

Step 5.2.2.1.4.1.23

Move $23$ to the left of $i$ .

$m = - \frac{27 - 27 i \sqrt{23} - 207 + 23 i \sqrt{23}}{8}$

Step 5.2.2.1.4.2

Simplify terms.

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Step 5.2.2.1.4.2.1

Subtract $207$ from $27$ .

$m = - \frac{- 27 i \sqrt{23} - 180 + 23 i \sqrt{23}}{8}$

Step 5.2.2.1.4.2.2

Add $- 27 i \sqrt{23}$ and $23 i \sqrt{23}$ .

$m = - \frac{- 4 i \sqrt{23} - 180}{8}$

Step 5.2.2.1.4.2.3

Reorder $- 4 i \sqrt{23}$ and $- 180$ .

$m = - \frac{- 180 - 4 i \sqrt{23}}{8}$

Step 5.2.2.1.4.2.4

Cancel the common factor of $- 180 - 4 i \sqrt{23}$ and $8$ .

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Step 5.2.2.1.4.2.4.1

Factor $4$ out of $- 180$ .

$m = - \frac{4 (- 45) - 4 i \sqrt{23}}{8}$

Step 5.2.2.1.4.2.4.2

Factor $4$ out of $- 4 i \sqrt{23}$ .

$m = - \frac{4 (- 45) + 4 (- i \sqrt{23})}{8}$

Step 5.2.2.1.4.2.4.3

Factor $4$ out of $4 (- 45) + 4 (- i \sqrt{23})$ .

$m = - \frac{4 (- 45 - i \sqrt{23})}{8}$

Step 5.2.2.1.4.2.4.4

Cancel the common factors.

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Step 5.2.2.1.4.2.4.4.1

Factor $4$ out of $8$ .

$m = - \frac{4 (- 45 - i \sqrt{23})}{4 \cdot 2}$

Step 5.2.2.1.4.2.4.4.2

Cancel the common factor.

$m = - \frac{4 (- 45 - i \sqrt{23})}{4 \cdot 2}$

Step 5.2.2.1.4.2.4.4.3

Rewrite the expression.

$m = - \frac{- 45 - i \sqrt{23}}{2}$

Step 5.2.2.1.4.2.5

Rewrite $- 45$ as $- 1 (45)$ .

$m = - \frac{- 1 (45) - i \sqrt{23}}{2}$

Step 5.2.2.1.4.2.6

Factor $- 1$ out of $- i \sqrt{23}$ .

$m = - \frac{- 1 (45) - (i \sqrt{23})}{2}$

Step 5.2.2.1.4.2.7

Factor $- 1$ out of $- 1 (45) - (i \sqrt{23})$ .

$m = - \frac{- 1 (45 + i \sqrt{23})}{2}$

Step 5.2.2.1.4.2.8

Simplify the expression.

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Step 5.2.2.1.4.2.8.1

Move the negative in front of the fraction.

$m = - - \frac{45 + i \sqrt{23}}{2}$

Step 5.2.2.1.4.2.8.2

Multiply $- 1$ by $- 1$ .

$m = 1 \frac{45 + i \sqrt{23}}{2}$

Step 5.2.2.1.4.2.8.3

Multiply $\frac{45 + i \sqrt{23}}{2}$ by $1$ .

$m = \frac{45 + i \sqrt{23}}{2}$

Step 6

Solve for $m^{\frac{1}{3}} = - \frac{3 + i \sqrt{23}}{2}$ for $m$ .

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Step 6.1

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

${(m^{\frac{1}{3}})}^{3} = {(- \frac{3 + i \sqrt{23}}{2})}^{3}$

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 ${(m^{\frac{1}{3}})}^{3}$ .

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Step 6.2.1.1.1

Multiply the exponents in ${(m^{\frac{1}{3}})}^{3}$ .

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Step 6.2.1.1.1.1

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

$m^{\frac{1}{3} \cdot 3} = {(- \frac{3 + i \sqrt{23}}{2})}^{3}$

Step 6.2.1.1.1.2

Cancel the common factor of $3$ .

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Step 6.2.1.1.1.2.1

Cancel the common factor.

$m^{\frac{1}{3} \cdot 3} = {(- \frac{3 + i \sqrt{23}}{2})}^{3}$

Step 6.2.1.1.1.2.2

Rewrite the expression.

$m^{1} = {(- \frac{3 + i \sqrt{23}}{2})}^{3}$

Step 6.2.1.1.2

Simplify.

$m = {(- \frac{3 + i \sqrt{23}}{2})}^{3}$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Simplify ${(- \frac{3 + i \sqrt{23}}{2})}^{3}$ .

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Step 6.2.2.1.1

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

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Step 6.2.2.1.1.1

Apply the product rule to $- \frac{3 + i \sqrt{23}}{2}$ .

$m = {(- 1)}^{3} {(\frac{3 + i \sqrt{23}}{2})}^{3}$

Step 6.2.2.1.1.2

Apply the product rule to $\frac{3 + i \sqrt{23}}{2}$ .

$m = {(- 1)}^{3} \frac{{(3 + i \sqrt{23})}^{3}}{2^{3}}$

Step 6.2.2.1.2

Evaluate the exponents.

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Step 6.2.2.1.2.1

Raise $- 1$ to the power of $3$ .

$m = - \frac{{(3 + i \sqrt{23})}^{3}}{2^{3}}$

Step 6.2.2.1.2.2

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

$m = - \frac{{(3 + i \sqrt{23})}^{3}}{8}$

Step 6.2.2.1.3

Use the Binomial Theorem.

$m = - \frac{3^{3} + 3 \cdot 3^{2} (i \sqrt{23}) + 3 \cdot 3 {(i \sqrt{23})}^{2} + {(i \sqrt{23})}^{3}}{8}$

Step 6.2.2.1.4

Simplify terms.

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Step 6.2.2.1.4.1

Simplify each term.

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Step 6.2.2.1.4.1.1

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

$m = - \frac{27 + 3 \cdot 3^{2} (i \sqrt{23}) + 3 \cdot 3 {(i \sqrt{23})}^{2} + {(i \sqrt{23})}^{3}}{8}$

Step 6.2.2.1.4.1.2

Multiply $3$ by $3^{2}$ by adding the exponents.

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Step 6.2.2.1.4.1.2.1

Multiply $3$ by $3^{2}$ .

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Step 6.2.2.1.4.1.2.1.1

Raise $3$ to the power of $1$ .

$m = - \frac{27 + 3^{1} \cdot 3^{2} (i \sqrt{23}) + 3 \cdot 3 {(i \sqrt{23})}^{2} + {(i \sqrt{23})}^{3}}{8}$

Step 6.2.2.1.4.1.2.1.2

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

$m = - \frac{27 + 3^{1 + 2} (i \sqrt{23}) + 3 \cdot 3 {(i \sqrt{23})}^{2} + {(i \sqrt{23})}^{3}}{8}$

Step 6.2.2.1.4.1.2.2

Add $1$ and $2$ .

$m = - \frac{27 + 3^{3} (i \sqrt{23}) + 3 \cdot 3 {(i \sqrt{23})}^{2} + {(i \sqrt{23})}^{3}}{8}$

Step 6.2.2.1.4.1.3

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

$m = - \frac{27 + 27 (i \sqrt{23}) + 3 \cdot 3 {(i \sqrt{23})}^{2} + {(i \sqrt{23})}^{3}}{8}$

Step 6.2.2.1.4.1.4

Multiply $3$ by $3$ .

$m = - \frac{27 + 27 i \sqrt{23} + 9 {(i \sqrt{23})}^{2} + {(i \sqrt{23})}^{3}}{8}$

Step 6.2.2.1.4.1.5

Apply the product rule to $i \sqrt{23}$ .

$m = - \frac{27 + 27 i \sqrt{23} + 9 (i^{2} {\sqrt{23}}^{2}) + {(i \sqrt{23})}^{3}}{8}$

Step 6.2.2.1.4.1.6

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

$m = - \frac{27 + 27 i \sqrt{23} + 9 (- 1 {\sqrt{23}}^{2}) + {(i \sqrt{23})}^{3}}{8}$

Step 6.2.2.1.4.1.7

Rewrite ${\sqrt{23}}^{2}$ as $23$ .

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Step 6.2.2.1.4.1.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{23}$ as $23^{\frac{1}{2}}$ .

$m = - \frac{27 + 27 i \sqrt{23} + 9 (- 1 {(23^{\frac{1}{2}})}^{2}) + {(i \sqrt{23})}^{3}}{8}$

Step 6.2.2.1.4.1.7.2

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

$m = - \frac{27 + 27 i \sqrt{23} + 9 (- 1 \cdot 23^{\frac{1}{2} \cdot 2}) + {(i \sqrt{23})}^{3}}{8}$

Step 6.2.2.1.4.1.7.3

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

$m = - \frac{27 + 27 i \sqrt{23} + 9 (- 1 \cdot 23^{\frac{2}{2}}) + {(i \sqrt{23})}^{3}}{8}$

Step 6.2.2.1.4.1.7.4

Cancel the common factor of $2$ .

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Step 6.2.2.1.4.1.7.4.1

Cancel the common factor.

$m = - \frac{27 + 27 i \sqrt{23} + 9 (- 1 \cdot 23^{\frac{2}{2}}) + {(i \sqrt{23})}^{3}}{8}$

Step 6.2.2.1.4.1.7.4.2

Rewrite the expression.

$m = - \frac{27 + 27 i \sqrt{23} + 9 (- 1 \cdot 23^{1}) + {(i \sqrt{23})}^{3}}{8}$

Step 6.2.2.1.4.1.7.5

Evaluate the exponent.

$m = - \frac{27 + 27 i \sqrt{23} + 9 (- 1 \cdot 23) + {(i \sqrt{23})}^{3}}{8}$

Step 6.2.2.1.4.1.8

Multiply $9 (- 1 \cdot 23)$ .

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Step 6.2.2.1.4.1.8.1

Multiply $- 1$ by $23$ .

$m = - \frac{27 + 27 i \sqrt{23} + 9 \cdot - 23 + {(i \sqrt{23})}^{3}}{8}$

Step 6.2.2.1.4.1.8.2

Multiply $9$ by $- 23$ .

$m = - \frac{27 + 27 i \sqrt{23} - 207 + {(i \sqrt{23})}^{3}}{8}$

Step 6.2.2.1.4.1.9

Apply the product rule to $i \sqrt{23}$ .

$m = - \frac{27 + 27 i \sqrt{23} - 207 + i^{3} {\sqrt{23}}^{3}}{8}$

Step 6.2.2.1.4.1.10

Factor out $i^{2}$ .

$m = - \frac{27 + 27 i \sqrt{23} - 207 + i^{2} \cdot i {\sqrt{23}}^{3}}{8}$

Step 6.2.2.1.4.1.11

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

$m = - \frac{27 + 27 i \sqrt{23} - 207 - 1 \cdot i {\sqrt{23}}^{3}}{8}$

Step 6.2.2.1.4.1.12

Rewrite $- 1 i$ as $- i$ .

$m = - \frac{27 + 27 i \sqrt{23} - 207 - i {\sqrt{23}}^{3}}{8}$

Step 6.2.2.1.4.1.13

Rewrite ${\sqrt{23}}^{3}$ as $\sqrt{23^{3}}$ .

$m = - \frac{27 + 27 i \sqrt{23} - 207 - i \sqrt{23^{3}}}{8}$

Step 6.2.2.1.4.1.14

Raise $23$ to the power of $3$ .

$m = - \frac{27 + 27 i \sqrt{23} - 207 - i \sqrt{12167}}{8}$

Step 6.2.2.1.4.1.15

Rewrite $12167$ as $23^{2} \cdot 23$ .

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Step 6.2.2.1.4.1.15.1

Factor $529$ out of $12167$ .

$m = - \frac{27 + 27 i \sqrt{23} - 207 - i \sqrt{529 (23)}}{8}$

Step 6.2.2.1.4.1.15.2

Rewrite $529$ as $23^{2}$ .

$m = - \frac{27 + 27 i \sqrt{23} - 207 - i \sqrt{23^{2} \cdot 23}}{8}$

Step 6.2.2.1.4.1.16

Pull terms out from under the radical.

$m = - \frac{27 + 27 i \sqrt{23} - 207 - i (23 \sqrt{23})}{8}$

Step 6.2.2.1.4.1.17

Multiply $23$ by $- 1$ .

$m = - \frac{27 + 27 i \sqrt{23} - 207 - 23 i \sqrt{23}}{8}$

Step 6.2.2.1.4.2

Simplify terms.

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Step 6.2.2.1.4.2.1

Subtract $207$ from $27$ .

$m = - \frac{27 i \sqrt{23} - 180 - 23 i \sqrt{23}}{8}$

Step 6.2.2.1.4.2.2

Subtract $23 i \sqrt{23}$ from $27 i \sqrt{23}$ .

$m = - \frac{4 i \sqrt{23} - 180}{8}$

Step 6.2.2.1.4.2.3

Reorder $4 i \sqrt{23}$ and $- 180$ .

$m = - \frac{- 180 + 4 i \sqrt{23}}{8}$

Step 6.2.2.1.4.2.4

Cancel the common factor of $- 180 + 4 i \sqrt{23}$ and $8$ .

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Step 6.2.2.1.4.2.4.1

Factor $4$ out of $- 180$ .

$m = - \frac{4 \cdot - 45 + 4 i \sqrt{23}}{8}$

Step 6.2.2.1.4.2.4.2

Factor $4$ out of $4 i \sqrt{23}$ .

$m = - \frac{4 \cdot - 45 + 4 (i \sqrt{23})}{8}$

Step 6.2.2.1.4.2.4.3

Factor $4$ out of $4 \cdot - 45 + 4 (i \sqrt{23})$ .

$m = - \frac{4 \cdot (- 45 + i \sqrt{23})}{8}$

Step 6.2.2.1.4.2.4.4

Cancel the common factors.

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Step 6.2.2.1.4.2.4.4.1

Factor $4$ out of $8$ .

$m = - \frac{4 \cdot (- 45 + i \sqrt{23})}{4 (2)}$

Step 6.2.2.1.4.2.4.4.2

Cancel the common factor.

$m = - \frac{4 \cdot (- 45 + i \sqrt{23})}{4 \cdot 2}$

Step 6.2.2.1.4.2.4.4.3

Rewrite the expression.

$m = - \frac{- 45 + i \sqrt{23}}{2}$

Step 6.2.2.1.4.2.5

Rewrite $- 45$ as $- 1 (45)$ .

$m = - \frac{- 1 (45) + i \sqrt{23}}{2}$

Step 6.2.2.1.4.2.6

Factor $- 1$ out of $i \sqrt{23}$ .

$m = - \frac{- 1 (45) - (- i \sqrt{23})}{2}$

Step 6.2.2.1.4.2.7

Factor $- 1$ out of $- 1 (45) - (- i \sqrt{23})$ .

$m = - \frac{- 1 (45 - i \sqrt{23})}{2}$

Step 6.2.2.1.4.2.8

Simplify the expression.

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Step 6.2.2.1.4.2.8.1

Move the negative in front of the fraction.

$m = - - \frac{45 - i \sqrt{23}}{2}$

Step 6.2.2.1.4.2.8.2

Multiply $- 1$ by $- 1$ .

$m = 1 \frac{45 - i \sqrt{23}}{2}$

Step 6.2.2.1.4.2.8.3

Multiply $\frac{45 - i \sqrt{23}}{2}$ by $1$ .

$m = \frac{45 - i \sqrt{23}}{2}$

Step 7

List all of the solutions.

$u = \frac{- 3 \pm i \sqrt{23}}{2}$