Algebra Examples

$x^{- 3} = \frac{1}{64}$

Step 1

Subtract $\frac{1}{64}$ from both sides of the equation.

$x^{- 3} - \frac{1}{64} = 0$

Step 2

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

$\frac{1}{x^{3}} - \frac{1}{64} = 0$

Step 3

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

$\frac{1^{3}}{x^{3}} - \frac{1}{64} = 0$

Step 4

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

${(\frac{1}{x})}^{3} - \frac{1}{64} = 0$

Step 5

Rewrite $\frac{1}{64}$ as ${(\frac{1}{4})}^{3}$ .

${(\frac{1}{x})}^{3} - {(\frac{1}{4})}^{3} = 0$

Step 6

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 $a = \frac{1}{x}$ and $b = \frac{1}{4}$ .

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

Step 7

Simplify.

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Step 7.1

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

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

Step 7.2

One to any power is one.

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

Step 7.3

Multiply $\frac{1}{x}$ by $\frac{1}{4}$ .

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

Step 7.4

Move $4$ to the left of $x$ .

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

Step 7.5

Multiply $4$ by $x$ .

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

Step 7.6

Apply the product rule to $\frac{1}{4}$ .

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

Step 7.7

One to any power is one.

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

Step 7.8

Raise $4$ to the power of $2$ .

$(\frac{1}{x} - \frac{1}{4}) (\frac{1}{x^{2}} + \frac{1}{4 x} + \frac{1}{16}) = 0$

Step 7.9

Reorder terms.

$(\frac{1}{x} - \frac{1}{4}) (\frac{1}{4 x} + \frac{1}{x^{2}} + \frac{1}{16}) = 0$

Step 8

To write $\frac{1}{x}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$(\frac{1}{x} \cdot \frac{4}{4} - \frac{1}{4}) (\frac{1}{4 x} + \frac{1}{x^{2}} + \frac{1}{16}) = 0$

Step 9

To write $- \frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$(\frac{1}{x} \cdot \frac{4}{4} - \frac{1}{4} \cdot \frac{x}{x}) (\frac{1}{4 x} + \frac{1}{x^{2}} + \frac{1}{16}) = 0$

Step 10

Write each expression with a common denominator of $x \cdot 4$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.1

Multiply $\frac{1}{x}$ by $\frac{4}{4}$ .

$(\frac{4}{x \cdot 4} - \frac{1}{4} \cdot \frac{x}{x}) (\frac{1}{4 x} + \frac{1}{x^{2}} + \frac{1}{16}) = 0$

Step 10.2

Multiply $\frac{1}{4}$ by $\frac{x}{x}$ .

$(\frac{4}{x \cdot 4} - \frac{x}{4 x}) (\frac{1}{4 x} + \frac{1}{x^{2}} + \frac{1}{16}) = 0$

Step 10.3

Reorder the factors of $x \cdot 4$ .

$(\frac{4}{4 x} - \frac{x}{4 x}) (\frac{1}{4 x} + \frac{1}{x^{2}} + \frac{1}{16}) = 0$

Step 11

Combine the numerators over the common denominator.

$\frac{4 - x}{4 x} \cdot (\frac{1}{4 x} + \frac{1}{x^{2}} + \frac{1}{16}) = 0$

Step 12

To write $\frac{1}{4 x}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{4 - x}{4 x} \cdot (\frac{1}{4 x} \cdot \frac{x}{x} + \frac{1}{x^{2}} + \frac{1}{16}) = 0$

Step 13

To write $\frac{1}{x^{2}}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{4 - x}{4 x} \cdot (\frac{1}{4 x} \cdot \frac{x}{x} + \frac{1}{x^{2}} \cdot \frac{4}{4} + \frac{1}{16}) = 0$

Step 14

Write each expression with a common denominator of $4 x^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 14.1

Multiply $\frac{1}{4 x}$ by $\frac{x}{x}$ .

$\frac{4 - x}{4 x} \cdot (\frac{x}{4 x \cdot x} + \frac{1}{x^{2}} \cdot \frac{4}{4} + \frac{1}{16}) = 0$

Step 14.2

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

$\frac{4 - x}{4 x} \cdot (\frac{x}{4 (x \cdot x)} + \frac{1}{x^{2}} \cdot \frac{4}{4} + \frac{1}{16}) = 0$

Step 14.3

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

$\frac{4 - x}{4 x} \cdot (\frac{x}{4 (x \cdot x)} + \frac{1}{x^{2}} \cdot \frac{4}{4} + \frac{1}{16}) = 0$

Step 14.4

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

$\frac{4 - x}{4 x} \cdot (\frac{x}{4 x^{1 + 1}} + \frac{1}{x^{2}} \cdot \frac{4}{4} + \frac{1}{16}) = 0$

Step 14.5

Add $1$ and $1$ .

$\frac{4 - x}{4 x} \cdot (\frac{x}{4 x^{2}} + \frac{1}{x^{2}} \cdot \frac{4}{4} + \frac{1}{16}) = 0$

Step 14.6

Multiply $\frac{1}{x^{2}}$ by $\frac{4}{4}$ .

$\frac{4 - x}{4 x} \cdot (\frac{x}{4 x^{2}} + \frac{4}{x^{2} \cdot 4} + \frac{1}{16}) = 0$

Step 14.7

Reorder the factors of $x^{2} \cdot 4$ .

$\frac{4 - x}{4 x} \cdot (\frac{x}{4 x^{2}} + \frac{4}{4 x^{2}} + \frac{1}{16}) = 0$

Step 15

Combine the numerators over the common denominator.

$\frac{4 - x}{4 x} \cdot (\frac{x + 4}{4 x^{2}} + \frac{1}{16}) = 0$

Step 16

To write $\frac{x + 4}{4 x^{2}}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{4 - x}{4 x} \cdot (\frac{x + 4}{4 x^{2}} \cdot \frac{4}{4} + \frac{1}{16}) = 0$

Step 17

To write $\frac{1}{16}$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$\frac{4 - x}{4 x} \cdot (\frac{x + 4}{4 x^{2}} \cdot \frac{4}{4} + \frac{1}{16} \cdot \frac{x^{2}}{x^{2}}) = 0$

Step 18

Write each expression with a common denominator of $16 x^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 18.1

Multiply $\frac{x + 4}{4 x^{2}}$ by $\frac{4}{4}$ .

$\frac{4 - x}{4 x} \cdot (\frac{(x + 4) \cdot 4}{4 x^{2} \cdot 4} + \frac{1}{16} \cdot \frac{x^{2}}{x^{2}}) = 0$

Step 18.2

Multiply $4$ by $4$ .

$\frac{4 - x}{4 x} \cdot (\frac{(x + 4) \cdot 4}{16 x^{2}} + \frac{1}{16} \cdot \frac{x^{2}}{x^{2}}) = 0$

Step 18.3

Multiply $\frac{1}{16}$ by $\frac{x^{2}}{x^{2}}$ .

$\frac{4 - x}{4 x} \cdot (\frac{(x + 4) \cdot 4}{16 x^{2}} + \frac{x^{2}}{16 x^{2}}) = 0$

Step 19

Combine the numerators over the common denominator.

$\frac{4 - x}{4 x} \cdot \frac{(x + 4) \cdot 4 + x^{2}}{16 x^{2}} = 0$

Step 20

Simplify the numerator.

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Step 20.1

Apply the distributive property.

$\frac{4 - x}{4 x} \cdot \frac{x \cdot 4 + 4 \cdot 4 + x^{2}}{16 x^{2}} = 0$

Step 20.2

Move $4$ to the left of $x$ .

$\frac{4 - x}{4 x} \cdot \frac{4 \cdot x + 4 \cdot 4 + x^{2}}{16 x^{2}} = 0$

Step 20.3

Multiply $4$ by $4$ .

$\frac{4 - x}{4 x} \cdot \frac{4 x + 16 + x^{2}}{16 x^{2}} = 0$

Step 20.4

Reorder terms.

$\frac{4 - x}{4 x} \cdot \frac{x^{2} + 4 x + 16}{16 x^{2}} = 0$

Step 21

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

$\frac{4 - x}{4 x} = 0$

$\frac{x^{2} + 4 x + 16}{16 x^{2}} = 0$

Step 22

Set $\frac{4 - x}{4 x}$ equal to $0$ and solve for $x$ .

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Step 22.1

Set $\frac{4 - x}{4 x}$ equal to $0$ .

$\frac{4 - x}{4 x} = 0$

Step 22.2

Solve $\frac{4 - x}{4 x} = 0$ for $x$ .

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Step 22.2.1

Set the numerator equal to zero.

$4 - x = 0$

Step 22.2.2

Solve the equation for $x$ .

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Step 22.2.2.1

Subtract $4$ from both sides of the equation.

$- x = - 4$

Step 22.2.2.2

Divide each term in $- x = - 4$ by $- 1$ and simplify.

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Step 22.2.2.2.1

Divide each term in $- x = - 4$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 4}{- 1}$

Step 22.2.2.2.2

Simplify the left side.

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Step 22.2.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 4}{- 1}$

Step 22.2.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 4}{- 1}$

Step 22.2.2.2.3

Simplify the right side.

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Step 22.2.2.2.3.1

Divide $- 4$ by $- 1$ .

$x = 4$

Step 23

Set $\frac{x^{2} + 4 x + 16}{16 x^{2}}$ equal to $0$ and solve for $x$ .

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Step 23.1

Set $\frac{x^{2} + 4 x + 16}{16 x^{2}}$ equal to $0$ .

$\frac{x^{2} + 4 x + 16}{16 x^{2}} = 0$

Step 23.2

Solve $\frac{x^{2} + 4 x + 16}{16 x^{2}} = 0$ for $x$ .

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Step 23.2.1

Set the numerator equal to zero.

$x^{2} + 4 x + 16 = 0$

Step 23.2.2

Solve the equation for $x$ .

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Step 23.2.2.1

Use the quadratic formula to find the solutions.

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

Step 23.2.2.2

Substitute the values $a = 1$ , $b = 4$ , and $c = 16$ into the quadratic formula and solve for $x$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (1 \cdot 16)}}{2 \cdot 1}$

Step 23.2.2.3

Simplify.

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Step 23.2.2.3.1

Simplify the numerator.

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Step 23.2.2.3.1.1

Raise $4$ to the power of $2$ .

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

Step 23.2.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 16$ .

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Step 23.2.2.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 23.2.2.3.1.2.2

Multiply $- 4$ by $16$ .

$x = \frac{- 4 \pm \sqrt{16 - 64}}{2 \cdot 1}$

Step 23.2.2.3.1.3

Subtract $64$ from $16$ .

$x = \frac{- 4 \pm \sqrt{- 48}}{2 \cdot 1}$

Step 23.2.2.3.1.4

Rewrite $- 48$ as $- 1 (48)$ .

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 48}}{2 \cdot 1}$

Step 23.2.2.3.1.5

Rewrite $\sqrt{- 1 (48)}$ as $\sqrt{- 1} \cdot \sqrt{48}$ .

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot 1}$

Step 23.2.2.3.1.6

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

$x = \frac{- 4 \pm i \cdot \sqrt{48}}{2 \cdot 1}$

Step 23.2.2.3.1.7

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 23.2.2.3.1.7.1

Factor $16$ out of $48$ .

$x = \frac{- 4 \pm i \cdot \sqrt{16 (3)}}{2 \cdot 1}$

Step 23.2.2.3.1.7.2

Rewrite $16$ as $4^{2}$ .

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

Step 23.2.2.3.1.8

Pull terms out from under the radical.

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

Step 23.2.2.3.1.9

Move $4$ to the left of $i$ .

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

Step 23.2.2.3.2

Multiply $2$ by $1$ .

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

Step 23.2.2.3.3

Simplify $\frac{- 4 \pm 4 i \sqrt{3}}{2}$ .

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

Step 23.2.2.4

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

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Step 23.2.2.4.1

Simplify the numerator.

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Step 23.2.2.4.1.1

Raise $4$ to the power of $2$ .

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

Step 23.2.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 16$ .

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Step 23.2.2.4.1.2.1

Multiply $- 4$ by $1$ .

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

Step 23.2.2.4.1.2.2

Multiply $- 4$ by $16$ .

$x = \frac{- 4 \pm \sqrt{16 - 64}}{2 \cdot 1}$

Step 23.2.2.4.1.3

Subtract $64$ from $16$ .

$x = \frac{- 4 \pm \sqrt{- 48}}{2 \cdot 1}$

Step 23.2.2.4.1.4

Rewrite $- 48$ as $- 1 (48)$ .

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 48}}{2 \cdot 1}$

Step 23.2.2.4.1.5

Rewrite $\sqrt{- 1 (48)}$ as $\sqrt{- 1} \cdot \sqrt{48}$ .

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot 1}$

Step 23.2.2.4.1.6

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

$x = \frac{- 4 \pm i \cdot \sqrt{48}}{2 \cdot 1}$

Step 23.2.2.4.1.7

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 23.2.2.4.1.7.1

Factor $16$ out of $48$ .

$x = \frac{- 4 \pm i \cdot \sqrt{16 (3)}}{2 \cdot 1}$

Step 23.2.2.4.1.7.2

Rewrite $16$ as $4^{2}$ .

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

Step 23.2.2.4.1.8

Pull terms out from under the radical.

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

Step 23.2.2.4.1.9

Move $4$ to the left of $i$ .

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

Step 23.2.2.4.2

Multiply $2$ by $1$ .

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

Step 23.2.2.4.3

Simplify $\frac{- 4 \pm 4 i \sqrt{3}}{2}$ .

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

Step 23.2.2.4.4

Change the $\pm$ to $+$ .

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

Step 23.2.2.5

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

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Step 23.2.2.5.1

Simplify the numerator.

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Step 23.2.2.5.1.1

Raise $4$ to the power of $2$ .

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

Step 23.2.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 16$ .

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Step 23.2.2.5.1.2.1

Multiply $- 4$ by $1$ .

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

Step 23.2.2.5.1.2.2

Multiply $- 4$ by $16$ .

$x = \frac{- 4 \pm \sqrt{16 - 64}}{2 \cdot 1}$

Step 23.2.2.5.1.3

Subtract $64$ from $16$ .

$x = \frac{- 4 \pm \sqrt{- 48}}{2 \cdot 1}$

Step 23.2.2.5.1.4

Rewrite $- 48$ as $- 1 (48)$ .

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 48}}{2 \cdot 1}$

Step 23.2.2.5.1.5

Rewrite $\sqrt{- 1 (48)}$ as $\sqrt{- 1} \cdot \sqrt{48}$ .

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot 1}$

Step 23.2.2.5.1.6

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

$x = \frac{- 4 \pm i \cdot \sqrt{48}}{2 \cdot 1}$

Step 23.2.2.5.1.7

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 23.2.2.5.1.7.1

Factor $16$ out of $48$ .

$x = \frac{- 4 \pm i \cdot \sqrt{16 (3)}}{2 \cdot 1}$

Step 23.2.2.5.1.7.2

Rewrite $16$ as $4^{2}$ .

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

Step 23.2.2.5.1.8

Pull terms out from under the radical.

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

Step 23.2.2.5.1.9

Move $4$ to the left of $i$ .

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

Step 23.2.2.5.2

Multiply $2$ by $1$ .

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

Step 23.2.2.5.3

Simplify $\frac{- 4 \pm 4 i \sqrt{3}}{2}$ .

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

Step 23.2.2.5.4

Change the $\pm$ to $-$ .

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

Step 23.2.2.6

The final answer is the combination of both solutions.

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

Step 24

The final solution is all the values that make $\frac{4 - x}{4 x} \cdot \frac{x^{2} + 4 x + 16}{16 x^{2}} = 0$ true.

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