Algebra Examples

$x^{\frac{1}{2}} + 3 x^{- \frac{1}{2}} = 10 x^{- \frac{3}{2}}$

Step 1

Find a common factor $x^{- 3}$ that is present in each term.

${(x^{- 3})}^{- \frac{1}{6}} + 3 {(x^{- 3})}^{\frac{1}{6}} - 10 {(x^{- 3})}^{\frac{1}{2}}$

Step 2

Substitute $u$ for $x^{- 3}$ .

${(u)}^{- \frac{1}{6}} + 3 {(u)}^{\frac{1}{6}} - 10 {(u)}^{\frac{1}{2}} = 0$

Step 3

Solve for $u$ .

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Step 3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{u^{\frac{1}{6}}} + 3 u^{\frac{1}{6}} - 10 u^{\frac{1}{2}} = 0$

Step 3.2

Find the LCD of the terms in the equation.

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Step 3.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$u^{\frac{1}{6}}, 1, 1, 1$

Step 3.2.2

The LCM of one and any expression is the expression.

$u^{\frac{1}{6}}$

Step 3.3

Multiply each term in $\frac{1}{u^{\frac{1}{6}}} + 3 u^{\frac{1}{6}} - 10 u^{\frac{1}{2}} = 0$ by $u^{\frac{1}{6}}$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in $\frac{1}{u^{\frac{1}{6}}} + 3 u^{\frac{1}{6}} - 10 u^{\frac{1}{2}} = 0$ by $u^{\frac{1}{6}}$ .

$\frac{1}{u^{\frac{1}{6}}} u^{\frac{1}{6}} + 3 u^{\frac{1}{6}} u^{\frac{1}{6}} - 10 u^{\frac{1}{2}} u^{\frac{1}{6}} = 0 u^{\frac{1}{6}}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Simplify each term.

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Step 3.3.2.1.1

Cancel the common factor of $u^{\frac{1}{6}}$ .

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Step 3.3.2.1.1.1

Cancel the common factor.

$\frac{1}{u^{\frac{1}{6}}} u^{\frac{1}{6}} + 3 u^{\frac{1}{6}} u^{\frac{1}{6}} - 10 u^{\frac{1}{2}} u^{\frac{1}{6}} = 0 u^{\frac{1}{6}}$

Step 3.3.2.1.1.2

Rewrite the expression.

$1 + 3 u^{\frac{1}{6}} u^{\frac{1}{6}} - 10 u^{\frac{1}{2}} u^{\frac{1}{6}} = 0 u^{\frac{1}{6}}$

Step 3.3.2.1.2

Multiply $u^{\frac{1}{6}}$ by $u^{\frac{1}{6}}$ by adding the exponents.

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Step 3.3.2.1.2.1

Move $u^{\frac{1}{6}}$ .

$1 + 3 (u^{\frac{1}{6}} u^{\frac{1}{6}}) - 10 u^{\frac{1}{2}} u^{\frac{1}{6}} = 0 u^{\frac{1}{6}}$

Step 3.3.2.1.2.2

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

$1 + 3 u^{\frac{1}{6} + \frac{1}{6}} - 10 u^{\frac{1}{2}} u^{\frac{1}{6}} = 0 u^{\frac{1}{6}}$

Step 3.3.2.1.2.3

Combine the numerators over the common denominator.

$1 + 3 u^{\frac{1 + 1}{6}} - 10 u^{\frac{1}{2}} u^{\frac{1}{6}} = 0 u^{\frac{1}{6}}$

Step 3.3.2.1.2.4

Add $1$ and $1$ .

$1 + 3 u^{\frac{2}{6}} - 10 u^{\frac{1}{2}} u^{\frac{1}{6}} = 0 u^{\frac{1}{6}}$

Step 3.3.2.1.2.5

Cancel the common factor of $2$ and $6$ .

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Step 3.3.2.1.2.5.1

Factor $2$ out of $2$ .

$1 + 3 u^{\frac{2 (1)}{6}} - 10 u^{\frac{1}{2}} u^{\frac{1}{6}} = 0 u^{\frac{1}{6}}$

Step 3.3.2.1.2.5.2

Cancel the common factors.

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Step 3.3.2.1.2.5.2.1

Factor $2$ out of $6$ .

$1 + 3 u^{\frac{2 \cdot 1}{2 \cdot 3}} - 10 u^{\frac{1}{2}} u^{\frac{1}{6}} = 0 u^{\frac{1}{6}}$

Step 3.3.2.1.2.5.2.2

Cancel the common factor.

$1 + 3 u^{\frac{2 \cdot 1}{2 \cdot 3}} - 10 u^{\frac{1}{2}} u^{\frac{1}{6}} = 0 u^{\frac{1}{6}}$

Step 3.3.2.1.2.5.2.3

Rewrite the expression.

$1 + 3 u^{\frac{1}{3}} - 10 u^{\frac{1}{2}} u^{\frac{1}{6}} = 0 u^{\frac{1}{6}}$

Step 3.3.2.1.3

Multiply $u^{\frac{1}{2}}$ by $u^{\frac{1}{6}}$ by adding the exponents.

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Step 3.3.2.1.3.1

Move $u^{\frac{1}{6}}$ .

$1 + 3 u^{\frac{1}{3}} - 10 (u^{\frac{1}{6}} u^{\frac{1}{2}}) = 0 u^{\frac{1}{6}}$

Step 3.3.2.1.3.2

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

$1 + 3 u^{\frac{1}{3}} - 10 u^{\frac{1}{6} + \frac{1}{2}} = 0 u^{\frac{1}{6}}$

Step 3.3.2.1.3.3

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$1 + 3 u^{\frac{1}{3}} - 10 u^{\frac{1}{6} + \frac{1}{2} \cdot \frac{3}{3}} = 0 u^{\frac{1}{6}}$

Step 3.3.2.1.3.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.2.1.3.4.1

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

$1 + 3 u^{\frac{1}{3}} - 10 u^{\frac{1}{6} + \frac{3}{2 \cdot 3}} = 0 u^{\frac{1}{6}}$

Step 3.3.2.1.3.4.2

Multiply $2$ by $3$ .

$1 + 3 u^{\frac{1}{3}} - 10 u^{\frac{1}{6} + \frac{3}{6}} = 0 u^{\frac{1}{6}}$

Step 3.3.2.1.3.5

Combine the numerators over the common denominator.

$1 + 3 u^{\frac{1}{3}} - 10 u^{\frac{1 + 3}{6}} = 0 u^{\frac{1}{6}}$

Step 3.3.2.1.3.6

Add $1$ and $3$ .

$1 + 3 u^{\frac{1}{3}} - 10 u^{\frac{4}{6}} = 0 u^{\frac{1}{6}}$

Step 3.3.2.1.3.7

Cancel the common factor of $4$ and $6$ .

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Step 3.3.2.1.3.7.1

Factor $2$ out of $4$ .

$1 + 3 u^{\frac{1}{3}} - 10 u^{\frac{2 (2)}{6}} = 0 u^{\frac{1}{6}}$

Step 3.3.2.1.3.7.2

Cancel the common factors.

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Step 3.3.2.1.3.7.2.1

Factor $2$ out of $6$ .

$1 + 3 u^{\frac{1}{3}} - 10 u^{\frac{2 \cdot 2}{2 \cdot 3}} = 0 u^{\frac{1}{6}}$

Step 3.3.2.1.3.7.2.2

Cancel the common factor.

$1 + 3 u^{\frac{1}{3}} - 10 u^{\frac{2 \cdot 2}{2 \cdot 3}} = 0 u^{\frac{1}{6}}$

Step 3.3.2.1.3.7.2.3

Rewrite the expression.

$1 + 3 u^{\frac{1}{3}} - 10 u^{\frac{2}{3}} = 0 u^{\frac{1}{6}}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Multiply $0$ by $u^{\frac{1}{6}}$ .

$1 + 3 u^{\frac{1}{3}} - 10 u^{\frac{2}{3}} = 0$

Step 3.4

Solve the equation.

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Step 3.4.1

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

$1 + 3 u^{\frac{1}{3}} - 10 {(u^{\frac{1}{3}})}^{2}$

Step 3.4.2

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

$1 + 3 (u) - 10 {(u)}^{2} = 0$

Step 3.4.3

Solve for $u$ .

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Step 3.4.3.1

Remove parentheses.

$1 + 3 u - 10 u^{2} = 0$

Step 3.4.3.2

Factor the left side of the equation.

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Step 3.4.3.2.1

Factor $- 1$ out of $1 + 3 u - 10 u^{2}$ .

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Step 3.4.3.2.1.1

Reorder the expression.

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Step 3.4.3.2.1.1.1

Move $1$ .

$3 u - 10 u^{2} + 1 = 0$

Step 3.4.3.2.1.1.2

Reorder $3 u$ and $- 10 u^{2}$ .

$- 10 u^{2} + 3 u + 1 = 0$

Step 3.4.3.2.1.2

Factor $- 1$ out of $- 10 u^{2}$ .

$- (10 u^{2}) + 3 u + 1 = 0$

Step 3.4.3.2.1.3

Factor $- 1$ out of $3 u$ .

$- (10 u^{2}) - (- 3 u) + 1 = 0$

Step 3.4.3.2.1.4

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

$- (10 u^{2}) - (- 3 u) - 1 \cdot - 1 = 0$

Step 3.4.3.2.1.5

Factor $- 1$ out of $- (10 u^{2}) - (- 3 u)$ .

$- (10 u^{2} - 3 u) - 1 \cdot - 1 = 0$

Step 3.4.3.2.1.6

Factor $- 1$ out of $- (10 u^{2} - 3 u) - 1 (- 1)$ .

$- (10 u^{2} - 3 u - 1) = 0$

Step 3.4.3.2.2

Factor.

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Step 3.4.3.2.2.1

Factor by grouping.

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Step 3.4.3.2.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 10 \cdot - 1 = - 10$ and whose sum is $b = - 3$ .

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Step 3.4.3.2.2.1.1.1

Factor $- 3$ out of $- 3 u$ .

$- (10 u^{2} - 3 u - 1) = 0$

Step 3.4.3.2.2.1.1.2

Rewrite $- 3$ as $2$ plus $- 5$

$- (10 u^{2} + (2 - 5) u - 1) = 0$

Step 3.4.3.2.2.1.1.3

Apply the distributive property.

$- (10 u^{2} + 2 u - 5 u - 1) = 0$

Step 3.4.3.2.2.1.2

Factor out the greatest common factor from each group.

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Step 3.4.3.2.2.1.2.1

Group the first two terms and the last two terms.

$- ((10 u^{2} + 2 u) - 5 u - 1) = 0$

Step 3.4.3.2.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$- (2 u (5 u + 1) - (5 u + 1)) = 0$

Step 3.4.3.2.2.1.3

Factor the polynomial by factoring out the greatest common factor, $5 u + 1$ .

$- ((5 u + 1) (2 u - 1)) = 0$

Step 3.4.3.2.2.2

Remove unnecessary parentheses.

$- (5 u + 1) (2 u - 1) = 0$

Step 3.4.3.3

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

$5 u + 1 = 0$

$2 u - 1 = 0$

Step 3.4.3.4

Set $5 u + 1$ equal to $0$ and solve for $u$ .

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Step 3.4.3.4.1

Set $5 u + 1$ equal to $0$ .

$5 u + 1 = 0$

Step 3.4.3.4.2

Solve $5 u + 1 = 0$ for $u$ .

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Step 3.4.3.4.2.1

Subtract $1$ from both sides of the equation.

$5 u = - 1$

Step 3.4.3.4.2.2

Divide each term in $5 u = - 1$ by $5$ and simplify.

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Step 3.4.3.4.2.2.1

Divide each term in $5 u = - 1$ by $5$ .

$\frac{5 u}{5} = \frac{- 1}{5}$

Step 3.4.3.4.2.2.2

Simplify the left side.

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Step 3.4.3.4.2.2.2.1

Cancel the common factor of $5$ .

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Step 3.4.3.4.2.2.2.1.1

Cancel the common factor.

$\frac{5 u}{5} = \frac{- 1}{5}$

Step 3.4.3.4.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{- 1}{5}$

Step 3.4.3.4.2.2.3

Simplify the right side.

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Step 3.4.3.4.2.2.3.1

Move the negative in front of the fraction.

$u = - \frac{1}{5}$

Step 3.4.3.5

Set $2 u - 1$ equal to $0$ and solve for $u$ .

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Step 3.4.3.5.1

Set $2 u - 1$ equal to $0$ .

$2 u - 1 = 0$

Step 3.4.3.5.2

Solve $2 u - 1 = 0$ for $u$ .

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Step 3.4.3.5.2.1

Add $1$ to both sides of the equation.

$2 u = 1$

Step 3.4.3.5.2.2

Divide each term in $2 u = 1$ by $2$ and simplify.

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Step 3.4.3.5.2.2.1

Divide each term in $2 u = 1$ by $2$ .

$\frac{2 u}{2} = \frac{1}{2}$

Step 3.4.3.5.2.2.2

Simplify the left side.

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Step 3.4.3.5.2.2.2.1

Cancel the common factor of $2$ .

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Step 3.4.3.5.2.2.2.1.1

Cancel the common factor.

$\frac{2 u}{2} = \frac{1}{2}$

Step 3.4.3.5.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{1}{2}$

Step 3.4.3.6

The final solution is all the values that make $- (5 u + 1) (2 u - 1) = 0$ true.

$u = - \frac{1}{5}, \frac{1}{2}$

Step 3.4.4

Substitute $u$ for $u$ .

$u^{\frac{1}{3}} = - \frac{1}{5}, \frac{1}{2}$

Step 3.4.5

Solve for $u^{\frac{1}{3}} = - \frac{1}{5}$ for $u$ .

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Step 3.4.5.1

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

${(u^{\frac{1}{3}})}^{3} = {(- \frac{1}{5})}^{3}$

Step 3.4.5.2

Simplify the exponent.

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Step 3.4.5.2.1

Simplify the left side.

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Step 3.4.5.2.1.1

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

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Step 3.4.5.2.1.1.1

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

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Step 3.4.5.2.1.1.1.1

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

$u^{\frac{1}{3} \cdot 3} = {(- \frac{1}{5})}^{3}$

Step 3.4.5.2.1.1.1.2

Cancel the common factor of $3$ .

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Step 3.4.5.2.1.1.1.2.1

Cancel the common factor.

$u^{\frac{1}{3} \cdot 3} = {(- \frac{1}{5})}^{3}$

Step 3.4.5.2.1.1.1.2.2

Rewrite the expression.

$u^{1} = {(- \frac{1}{5})}^{3}$

Step 3.4.5.2.1.1.2

Simplify.

$u = {(- \frac{1}{5})}^{3}$

Step 3.4.5.2.2

Simplify the right side.

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Step 3.4.5.2.2.1

Simplify ${(- \frac{1}{5})}^{3}$ .

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Step 3.4.5.2.2.1.1

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

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Step 3.4.5.2.2.1.1.1

Apply the product rule to $- \frac{1}{5}$ .

$u = {(- 1)}^{3} {(\frac{1}{5})}^{3}$

Step 3.4.5.2.2.1.1.2

Apply the product rule to $\frac{1}{5}$ .

$u = {(- 1)}^{3} \frac{1^{3}}{5^{3}}$

Step 3.4.5.2.2.1.2

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

$u = - \frac{1^{3}}{5^{3}}$

Step 3.4.5.2.2.1.3

One to any power is one.

$u = - \frac{1}{5^{3}}$

Step 3.4.5.2.2.1.4

Raise $5$ to the power of $3$ .

$u = - \frac{1}{125}$

Step 3.4.6

Solve for $u^{\frac{1}{3}} = \frac{1}{2}$ for $u$ .

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Step 3.4.6.1

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

${(u^{\frac{1}{3}})}^{3} = {(\frac{1}{2})}^{3}$

Step 3.4.6.2

Simplify the exponent.

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Step 3.4.6.2.1

Simplify the left side.

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Step 3.4.6.2.1.1

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

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Step 3.4.6.2.1.1.1

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

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Step 3.4.6.2.1.1.1.1

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

$u^{\frac{1}{3} \cdot 3} = {(\frac{1}{2})}^{3}$

Step 3.4.6.2.1.1.1.2

Cancel the common factor of $3$ .

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Step 3.4.6.2.1.1.1.2.1

Cancel the common factor.

$u^{\frac{1}{3} \cdot 3} = {(\frac{1}{2})}^{3}$

Step 3.4.6.2.1.1.1.2.2

Rewrite the expression.

$u^{1} = {(\frac{1}{2})}^{3}$

Step 3.4.6.2.1.1.2

Simplify.

$u = {(\frac{1}{2})}^{3}$

Step 3.4.6.2.2

Simplify the right side.

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Step 3.4.6.2.2.1

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

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Step 3.4.6.2.2.1.1

Apply the product rule to $\frac{1}{2}$ .

$u = \frac{1^{3}}{2^{3}}$

Step 3.4.6.2.2.1.2

One to any power is one.

$u = \frac{1}{2^{3}}$

Step 3.4.6.2.2.1.3

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

$u = \frac{1}{8}$

Step 3.4.7

List all of the solutions.

$u = - \frac{1}{125}, \frac{1}{8}$

Step 4

Substitute $x$ for $u$ .

$x^{- 3} = - \frac{1}{125}, \frac{1}{8}$

Step 5

Solve for $x^{- 3} = - \frac{1}{125}$ for $x$ .

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Step 5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{x^{3}} = - \frac{1}{125}$

Step 5.2

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$1 \cdot 125 = x^{3} \cdot - 1$

Step 5.3

Solve the equation for $x$ .

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Step 5.3.1

Rewrite the equation as $x^{3} \cdot - 1 = 1 \cdot 125$ .

$x^{3} \cdot - 1 = 1 \cdot 125$

Step 5.3.2

Divide each term in $x^{3} \cdot - 1 = 1 \cdot 125$ by $- 1$ and simplify.

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Step 5.3.2.1

Divide each term in $x^{3} \cdot - 1 = 1 \cdot 125$ by $- 1$ .

$\frac{x^{3} \cdot - 1}{- 1} = \frac{1 \cdot 125}{- 1}$

Step 5.3.2.2

Simplify the left side.

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Step 5.3.2.2.1

Move the negative one from the denominator of $\frac{x^{3} \cdot - 1}{- 1}$ .

$- 1 \cdot (x^{3} \cdot - 1) = \frac{1 \cdot 125}{- 1}$

Step 5.3.2.2.2

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

$- (x^{3} \cdot - 1) = \frac{1 \cdot 125}{- 1}$

Step 5.3.2.2.3

Move $- 1$ to the left of $x^{3}$ .

$- (- 1 \cdot x^{3}) = \frac{1 \cdot 125}{- 1}$

Step 5.3.2.2.4

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

$- - x^{3} = \frac{1 \cdot 125}{- 1}$

Step 5.3.2.2.5

Multiply $- - x^{3}$ .

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Step 5.3.2.2.5.1

Multiply $- 1$ by $- 1$ .

$1 x^{3} = \frac{1 \cdot 125}{- 1}$

Step 5.3.2.2.5.2

Multiply $x^{3}$ by $1$ .

$x^{3} = \frac{1 \cdot 125}{- 1}$

Step 5.3.2.3

Simplify the right side.

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Step 5.3.2.3.1

Move the negative one from the denominator of $\frac{1 \cdot 125}{- 1}$ .

$x^{3} = - 1 \cdot (1 \cdot 125)$

Step 5.3.2.3.2

Rewrite $- 1 \cdot (1 \cdot 125)$ as $- (1 \cdot 125)$ .

$x^{3} = - (1 \cdot 125)$

Step 5.3.2.3.3

Multiply $- (1 \cdot 125)$ .

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Step 5.3.2.3.3.1

Multiply $125$ by $1$ .

$x^{3} = - 1 \cdot 125$

Step 5.3.2.3.3.2

Multiply $- 1$ by $125$ .

$x^{3} = - 125$

Step 5.3.3

Add $125$ to both sides of the equation.

$x^{3} + 125 = 0$

Step 5.3.4

Factor the left side of the equation.

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Step 5.3.4.1

Rewrite $125$ as $5^{3}$ .

$x^{3} + 5^{3} = 0$

Step 5.3.4.2

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x$ and $b = 5$ .

$(x + 5) (x^{2} - x \cdot 5 + 5^{2}) = 0$

Step 5.3.4.3

Simplify.

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Step 5.3.4.3.1

Multiply $5$ by $- 1$ .

$(x + 5) (x^{2} - 5 x + 5^{2}) = 0$

Step 5.3.4.3.2

Raise $5$ to the power of $2$ .

$(x + 5) (x^{2} - 5 x + 25) = 0$

Step 5.3.5

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

$x + 5 = 0$

$x^{2} - 5 x + 25 = 0$

Step 5.3.6

Set $x + 5$ equal to $0$ and solve for $x$ .

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Step 5.3.6.1

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Step 5.3.6.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 5.3.7

Set $x^{2} - 5 x + 25$ equal to $0$ and solve for $x$ .

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Step 5.3.7.1

Set $x^{2} - 5 x + 25$ equal to $0$ .

$x^{2} - 5 x + 25 = 0$

Step 5.3.7.2

Solve $x^{2} - 5 x + 25 = 0$ for $x$ .

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Step 5.3.7.2.1

Use the quadratic formula to find the solutions.

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

Step 5.3.7.2.2

Substitute the values $a = 1$ , $b = - 5$ , and $c = 25$ into the quadratic formula and solve for $x$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (1 \cdot 25)}}{2 \cdot 1}$

Step 5.3.7.2.3

Simplify.

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Step 5.3.7.2.3.1

Simplify the numerator.

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Step 5.3.7.2.3.1.1

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 1 \cdot 25}}{2 \cdot 1}$

Step 5.3.7.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 25$ .

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Step 5.3.7.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 25}}{2 \cdot 1}$

Step 5.3.7.2.3.1.2.2

Multiply $- 4$ by $25$ .

$x = \frac{5 \pm \sqrt{25 - 100}}{2 \cdot 1}$

Step 5.3.7.2.3.1.3

Subtract $100$ from $25$ .

$x = \frac{5 \pm \sqrt{- 75}}{2 \cdot 1}$

Step 5.3.7.2.3.1.4

Rewrite $- 75$ as $- 1 (75)$ .

$x = \frac{5 \pm \sqrt{- 1 \cdot 75}}{2 \cdot 1}$

Step 5.3.7.2.3.1.5

Rewrite $\sqrt{- 1 (75)}$ as $\sqrt{- 1} \cdot \sqrt{75}$ .

$x = \frac{5 \pm \sqrt{- 1} \cdot \sqrt{75}}{2 \cdot 1}$

Step 5.3.7.2.3.1.6

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

$x = \frac{5 \pm i \cdot \sqrt{75}}{2 \cdot 1}$

Step 5.3.7.2.3.1.7

Rewrite $75$ as $5^{2} \cdot 3$ .

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Step 5.3.7.2.3.1.7.1

Factor $25$ out of $75$ .

$x = \frac{5 \pm i \cdot \sqrt{25 (3)}}{2 \cdot 1}$

Step 5.3.7.2.3.1.7.2

Rewrite $25$ as $5^{2}$ .

$x = \frac{5 \pm i \cdot \sqrt{5^{2} \cdot 3}}{2 \cdot 1}$

Step 5.3.7.2.3.1.8

Pull terms out from under the radical.

$x = \frac{5 \pm i \cdot (5 \sqrt{3})}{2 \cdot 1}$

Step 5.3.7.2.3.1.9

Move $5$ to the left of $i$ .

$x = \frac{5 \pm 5 i \sqrt{3}}{2 \cdot 1}$

Step 5.3.7.2.3.2

Multiply $2$ by $1$ .

$x = \frac{5 \pm 5 i \sqrt{3}}{2}$

Step 5.3.7.2.4

The final answer is the combination of both solutions.

$x = \frac{5 + 5 i \sqrt{3}}{2}, \frac{5 - 5 i \sqrt{3}}{2}$

Step 5.3.8

The final solution is all the values that make $(x + 5) (x^{2} - 5 x + 25) = 0$ true.

$x = - 5, \frac{5 + 5 i \sqrt{3}}{2}, \frac{5 - 5 i \sqrt{3}}{2}$

Step 6

Solve for $x^{- 3} = \frac{1}{8}$ for $x$ .

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Step 6.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{x^{3}} = \frac{1}{8}$

Step 6.2

$1 \cdot 8 = x^{3} \cdot 1$

Step 6.3

Solve the equation for $x$ .

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Step 6.3.1

Rewrite the equation as $x^{3} \cdot 1 = 1 \cdot 8$ .

$x^{3} \cdot 1 = 1 \cdot 8$

Step 6.3.2

Multiply $x^{3}$ by $1$ .

$x^{3} = 1 \cdot 8$

Step 6.3.3

Multiply $8$ by $1$ .

$x^{3} = 8$

Step 6.3.4

Subtract $8$ from both sides of the equation.

$x^{3} - 8 = 0$

Step 6.3.5

Factor the left side of the equation.

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Step 6.3.5.1

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

$x^{3} - 2^{3} = 0$

Step 6.3.5.2

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 $a = x$ and $b = 2$ .

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

Step 6.3.5.3

Simplify.

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Step 6.3.5.3.1

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

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

Step 6.3.5.3.2

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

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

Step 6.3.6

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

$x - 2 = 0$

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

Step 6.3.7

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 6.3.7.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 6.3.7.2

Add $2$ to both sides of the equation.

$x = 2$

Step 6.3.8

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

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Step 6.3.8.1

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

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

Step 6.3.8.2

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

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Step 6.3.8.2.1

Use the quadratic formula to find the solutions.

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

Step 6.3.8.2.2

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

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

Step 6.3.8.2.3

Simplify.

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Step 6.3.8.2.3.1

Simplify the numerator.

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Step 6.3.8.2.3.1.1

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

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

Step 6.3.8.2.3.1.2

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

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Step 6.3.8.2.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 6.3.8.2.3.1.2.2

Multiply $- 4$ by $4$ .

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

Step 6.3.8.2.3.1.3

Subtract $16$ from $4$ .

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

Step 6.3.8.2.3.1.4

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

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

Step 6.3.8.2.3.1.5

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

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

Step 6.3.8.2.3.1.6

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

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

Step 6.3.8.2.3.1.7

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

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Step 6.3.8.2.3.1.7.1

Factor $4$ out of $12$ .

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

Step 6.3.8.2.3.1.7.2

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

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

Step 6.3.8.2.3.1.8

Pull terms out from under the radical.

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

Step 6.3.8.2.3.1.9

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

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

Step 6.3.8.2.3.2

Multiply $2$ by $1$ .

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

Step 6.3.8.2.3.3

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

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

Step 6.3.8.2.4

The final answer is the combination of both solutions.

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

Step 6.3.9

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

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

Step 7

List all of the solutions.

$x = - 5, \frac{5 + 5 i \sqrt{3}}{2}, \frac{5 - 5 i \sqrt{3}}{2}, 2, - 1 + i \sqrt{3}, - 1 - i \sqrt{3}$