Algebra Examples

$x^{- 3} = 64$

Step 1

Subtract $64$ from both sides of the equation.

$x^{- 3} - 64 = 0$

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Rewrite $64$ as $4^{3}$ .

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

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 $b = 4$ .

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

Step 7

Simplify.

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Step 7.1

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

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

Step 7.2

One to any power is one.

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

Step 7.3

Combine $\frac{1}{x}$ and $4$ .

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

Step 7.4

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

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

Step 7.5

Reorder terms.

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

Step 8

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

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

Step 9

Combine $- 4$ and $\frac{x}{x}$ .

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

Step 10

Combine the numerators over the common denominator.

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

Step 11

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

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

Step 12

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

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Step 12.1

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

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

Step 12.2

Multiply $x$ by $x$ .

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

Step 13

Combine the numerators over the common denominator.

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

Step 14

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

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

Step 15

Combine the numerators over the common denominator.

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

Step 16

Reorder terms.

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

Step 17

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

$\frac{1 - 4 x}{x} = 0$

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

Step 18

Set $\frac{1 - 4 x}{x}$ equal to $0$ and solve for $x$ .

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Step 18.1

Set $\frac{1 - 4 x}{x}$ equal to $0$ .

$\frac{1 - 4 x}{x} = 0$

Step 18.2

Solve $\frac{1 - 4 x}{x} = 0$ for $x$ .

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Step 18.2.1

Set the numerator equal to zero.

$1 - 4 x = 0$

Step 18.2.2

Solve the equation for $x$ .

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Step 18.2.2.1

Subtract $1$ from both sides of the equation.

$- 4 x = - 1$

Step 18.2.2.2

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

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Step 18.2.2.2.1

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

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

Step 18.2.2.2.2

Simplify the left side.

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Step 18.2.2.2.2.1

Cancel the common factor of $- 4$ .

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Step 18.2.2.2.2.1.1

Cancel the common factor.

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

Step 18.2.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 18.2.2.2.3

Simplify the right side.

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Step 18.2.2.2.3.1

Dividing two negative values results in a positive value.

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

Step 19

Set $\frac{16 x^{2} + 4 x + 1}{x^{2}}$ equal to $0$ and solve for $x$ .

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Step 19.1

Set $\frac{16 x^{2} + 4 x + 1}{x^{2}}$ equal to $0$ .

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

Step 19.2

Solve $\frac{16 x^{2} + 4 x + 1}{x^{2}} = 0$ for $x$ .

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Step 19.2.1

Set the numerator equal to zero.

$16 x^{2} + 4 x + 1 = 0$

Step 19.2.2

Solve the equation for $x$ .

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Step 19.2.2.1

Use the quadratic formula to find the solutions.

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

Step 19.2.2.2

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

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

Step 19.2.2.3

Simplify.

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Step 19.2.2.3.1

Simplify the numerator.

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Step 19.2.2.3.1.1

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

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

Step 19.2.2.3.1.2

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

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Step 19.2.2.3.1.2.1

Multiply $- 4$ by $16$ .

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

Step 19.2.2.3.1.2.2

Multiply $- 64$ by $1$ .

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

Step 19.2.2.3.1.3

Subtract $64$ from $16$ .

$x = \frac{- 4 \pm \sqrt{- 48}}{2 \cdot 16}$

Step 19.2.2.3.1.4

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

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 48}}{2 \cdot 16}$

Step 19.2.2.3.1.5

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

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot 16}$

Step 19.2.2.3.1.6

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

$x = \frac{- 4 \pm i \cdot \sqrt{48}}{2 \cdot 16}$

Step 19.2.2.3.1.7

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

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Step 19.2.2.3.1.7.1

Factor $16$ out of $48$ .

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

Step 19.2.2.3.1.7.2

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

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

Step 19.2.2.3.1.8

Pull terms out from under the radical.

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

Step 19.2.2.3.1.9

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

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

Step 19.2.2.3.2

Multiply $2$ by $16$ .

$x = \frac{- 4 \pm 4 i \sqrt{3}}{32}$

Step 19.2.2.3.3

Simplify $\frac{- 4 \pm 4 i \sqrt{3}}{32}$ .

$x = \frac{- 1 \pm i \sqrt{3}}{8}$

Step 19.2.2.4

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

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Step 19.2.2.4.1

Simplify the numerator.

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Step 19.2.2.4.1.1

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

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

Step 19.2.2.4.1.2

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

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Step 19.2.2.4.1.2.1

Multiply $- 4$ by $16$ .

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

Step 19.2.2.4.1.2.2

Multiply $- 64$ by $1$ .

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

Step 19.2.2.4.1.3

Subtract $64$ from $16$ .

$x = \frac{- 4 \pm \sqrt{- 48}}{2 \cdot 16}$

Step 19.2.2.4.1.4

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

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 48}}{2 \cdot 16}$

Step 19.2.2.4.1.5

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

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot 16}$

Step 19.2.2.4.1.6

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

$x = \frac{- 4 \pm i \cdot \sqrt{48}}{2 \cdot 16}$

Step 19.2.2.4.1.7

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

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Step 19.2.2.4.1.7.1

Factor $16$ out of $48$ .

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

Step 19.2.2.4.1.7.2

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

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

Step 19.2.2.4.1.8

Pull terms out from under the radical.

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

Step 19.2.2.4.1.9

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

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

Step 19.2.2.4.2

Multiply $2$ by $16$ .

$x = \frac{- 4 \pm 4 i \sqrt{3}}{32}$

Step 19.2.2.4.3

Simplify $\frac{- 4 \pm 4 i \sqrt{3}}{32}$ .

$x = \frac{- 1 \pm i \sqrt{3}}{8}$

Step 19.2.2.4.4

Change the $\pm$ to $+$ .

$x = \frac{- 1 + i \sqrt{3}}{8}$

Step 19.2.2.4.5

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

$x = \frac{- 1 \cdot 1 + i \sqrt{3}}{8}$

Step 19.2.2.4.6

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

$x = \frac{- 1 \cdot 1 - (- i \sqrt{3})}{8}$

Step 19.2.2.4.7

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

$x = \frac{- 1 (1 - i \sqrt{3})}{8}$

Step 19.2.2.4.8

Move the negative in front of the fraction.

$x = - \frac{1 - i \sqrt{3}}{8}$

Step 19.2.2.5

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

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Step 19.2.2.5.1

Simplify the numerator.

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Step 19.2.2.5.1.1

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

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

Step 19.2.2.5.1.2

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

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Step 19.2.2.5.1.2.1

Multiply $- 4$ by $16$ .

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

Step 19.2.2.5.1.2.2

Multiply $- 64$ by $1$ .

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

Step 19.2.2.5.1.3

Subtract $64$ from $16$ .

$x = \frac{- 4 \pm \sqrt{- 48}}{2 \cdot 16}$

Step 19.2.2.5.1.4

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

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 48}}{2 \cdot 16}$

Step 19.2.2.5.1.5

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

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot 16}$

Step 19.2.2.5.1.6

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

$x = \frac{- 4 \pm i \cdot \sqrt{48}}{2 \cdot 16}$

Step 19.2.2.5.1.7

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

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Step 19.2.2.5.1.7.1

Factor $16$ out of $48$ .

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

Step 19.2.2.5.1.7.2

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

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

Step 19.2.2.5.1.8

Pull terms out from under the radical.

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

Step 19.2.2.5.1.9

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

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

Step 19.2.2.5.2

Multiply $2$ by $16$ .

$x = \frac{- 4 \pm 4 i \sqrt{3}}{32}$

Step 19.2.2.5.3

Simplify $\frac{- 4 \pm 4 i \sqrt{3}}{32}$ .

$x = \frac{- 1 \pm i \sqrt{3}}{8}$

Step 19.2.2.5.4

Change the $\pm$ to $-$ .

$x = \frac{- 1 - i \sqrt{3}}{8}$

Step 19.2.2.5.5

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

$x = \frac{- 1 \cdot 1 - i \sqrt{3}}{8}$

Step 19.2.2.5.6

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

$x = \frac{- 1 \cdot 1 - (i \sqrt{3})}{8}$

Step 19.2.2.5.7

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

$x = \frac{- 1 (1 + i \sqrt{3})}{8}$

Step 19.2.2.5.8

Move the negative in front of the fraction.

$x = - \frac{1 + i \sqrt{3}}{8}$

Step 19.2.2.6

The final answer is the combination of both solutions.

$x = - \frac{1 - i \sqrt{3}}{8}, - \frac{1 + i \sqrt{3}}{8}$

Step 20

The final solution is all the values that make $\frac{1 - 4 x}{x} \cdot \frac{16 x^{2} + 4 x + 1}{x^{2}} = 0$ true.

$x = \frac{1}{4}, - \frac{1 - i \sqrt{3}}{8}, - \frac{1 + i \sqrt{3}}{8}$