Precalculus Examples

${(8 x)}^{- 3} = 64$

Step 1

Simplify ${(8 x)}^{- 3}$ .

Tap for more steps...

Step 1.1

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

$\frac{1}{{(8 x)}^{3}} = 64$

Step 1.2

Simplify the denominator.

Tap for more steps...

Step 1.2.1

Apply the product rule to $8 x$ .

$\frac{1}{8^{3} x^{3}} = 64$

Step 1.2.2

Raise $8$ to the power of $3$ .

$\frac{1}{512 x^{3}} = 64$

Step 2

Find the LCD of the terms in the equation.

Tap for more steps...

Step 2.1

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

$512 x^{3}, 1$

Step 2.2

The LCM of one and any expression is the expression.

$512 x^{3}$

Step 3

Multiply each term in $\frac{1}{512 x^{3}} = 64$ by $512 x^{3}$ to eliminate the fractions.

Tap for more steps...

Step 3.1

Multiply each term in $\frac{1}{512 x^{3}} = 64$ by $512 x^{3}$ .

$\frac{1}{512 x^{3}} (512 x^{3}) = 64 (512 x^{3})$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Rewrite using the commutative property of multiplication.

$512 \frac{1}{512 x^{3}} x^{3} = 64 (512 x^{3})$

Step 3.2.2

Cancel the common factor of $512$ .

Tap for more steps...

Step 3.2.2.1

Factor $512$ out of $512 x^{3}$ .

$512 \frac{1}{512 (x^{3})} x^{3} = 64 (512 x^{3})$

Step 3.2.2.2

Cancel the common factor.

$512 \frac{1}{512 x^{3}} x^{3} = 64 (512 x^{3})$

Step 3.2.2.3

Rewrite the expression.

$\frac{1}{x^{3}} x^{3} = 64 (512 x^{3})$

Step 3.2.3

Cancel the common factor of $x^{3}$ .

Tap for more steps...

Step 3.2.3.1

Cancel the common factor.

$\frac{1}{x^{3}} x^{3} = 64 (512 x^{3})$

Step 3.2.3.2

Rewrite the expression.

$1 = 64 (512 x^{3})$

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Multiply $512$ by $64$ .

$1 = 32768 x^{3}$

Step 4

Solve the equation.

Tap for more steps...

Step 4.1

Rewrite the equation as $32768 x^{3} = 1$ .

$32768 x^{3} = 1$

Step 4.2

Subtract $1$ from both sides of the equation.

$32768 x^{3} - 1 = 0$

Step 4.3

Factor the left side of the equation.

Tap for more steps...

Step 4.3.1

Rewrite $32768 x^{3}$ as ${(32 x)}^{3}$ .

${(32 x)}^{3} - 1 = 0$

Step 4.3.2

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

${(32 x)}^{3} - 1^{3} = 0$

Step 4.3.3

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 $a = 32 x$ and $b = 1$ .

$(32 x - 1) ({(32 x)}^{2} + 32 x \cdot 1 + 1^{2}) = 0$

Step 4.3.4

Simplify.

Tap for more steps...

Step 4.3.4.1

Apply the product rule to $32 x$ .

$(32 x - 1) (32^{2} x^{2} + 32 x \cdot 1 + 1^{2}) = 0$

Step 4.3.4.2

Raise $32$ to the power of $2$ .

$(32 x - 1) (1024 x^{2} + 32 x \cdot 1 + 1^{2}) = 0$

Step 4.3.4.3

Multiply $32$ by $1$ .

$(32 x - 1) (1024 x^{2} + 32 x + 1^{2}) = 0$

Step 4.3.4.4

One to any power is one.

$(32 x - 1) (1024 x^{2} + 32 x + 1) = 0$

Step 4.4

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

$32 x - 1 = 0$

$1024 x^{2} + 32 x + 1 = 0$

Step 4.5

Set $32 x - 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.5.1

Set $32 x - 1$ equal to $0$ .

$32 x - 1 = 0$

Step 4.5.2

Solve $32 x - 1 = 0$ for $x$ .

Tap for more steps...

Step 4.5.2.1

Add $1$ to both sides of the equation.

$32 x = 1$

Step 4.5.2.2

Divide each term in $32 x = 1$ by $32$ and simplify.

Tap for more steps...

Step 4.5.2.2.1

Divide each term in $32 x = 1$ by $32$ .

$\frac{32 x}{32} = \frac{1}{32}$

Step 4.5.2.2.2

Simplify the left side.

Tap for more steps...

Step 4.5.2.2.2.1

Cancel the common factor of $32$ .

Tap for more steps...

Step 4.5.2.2.2.1.1

Cancel the common factor.

$\frac{32 x}{32} = \frac{1}{32}$

Step 4.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{32}$

Step 4.6

Set $1024 x^{2} + 32 x + 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.6.1

Set $1024 x^{2} + 32 x + 1$ equal to $0$ .

$1024 x^{2} + 32 x + 1 = 0$

Step 4.6.2

Solve $1024 x^{2} + 32 x + 1 = 0$ for $x$ .

Tap for more steps...

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 $a = 1024$ , $b = 32$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{- 32 \pm \sqrt{32^{2} - 4 \cdot (1024 \cdot 1)}}{2 \cdot 1024}$

Step 4.6.2.3

Simplify.

Tap for more steps...

Step 4.6.2.3.1

Simplify the numerator.

Tap for more steps...

Step 4.6.2.3.1.1

Raise $32$ to the power of $2$ .

$x = \frac{- 32 \pm \sqrt{1024 - 4 \cdot 1024 \cdot 1}}{2 \cdot 1024}$

Step 4.6.2.3.1.2

Multiply $- 4 \cdot 1024 \cdot 1$ .

Tap for more steps...

Step 4.6.2.3.1.2.1

Multiply $- 4$ by $1024$ .

$x = \frac{- 32 \pm \sqrt{1024 - 4096 \cdot 1}}{2 \cdot 1024}$

Step 4.6.2.3.1.2.2

Multiply $- 4096$ by $1$ .

$x = \frac{- 32 \pm \sqrt{1024 - 4096}}{2 \cdot 1024}$

Step 4.6.2.3.1.3

Subtract $4096$ from $1024$ .

$x = \frac{- 32 \pm \sqrt{- 3072}}{2 \cdot 1024}$

Step 4.6.2.3.1.4

Rewrite $- 3072$ as $- 1 (3072)$ .

$x = \frac{- 32 \pm \sqrt{- 1 \cdot 3072}}{2 \cdot 1024}$

Step 4.6.2.3.1.5

Rewrite $\sqrt{- 1 (3072)}$ as $\sqrt{- 1} \cdot \sqrt{3072}$ .

$x = \frac{- 32 \pm \sqrt{- 1} \cdot \sqrt{3072}}{2 \cdot 1024}$

Step 4.6.2.3.1.6

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

$x = \frac{- 32 \pm i \cdot \sqrt{3072}}{2 \cdot 1024}$

Step 4.6.2.3.1.7

Rewrite $3072$ as $32^{2} \cdot 3$ .

Tap for more steps...

Step 4.6.2.3.1.7.1

Factor $1024$ out of $3072$ .

$x = \frac{- 32 \pm i \cdot \sqrt{1024 (3)}}{2 \cdot 1024}$

Step 4.6.2.3.1.7.2

Rewrite $1024$ as $32^{2}$ .

$x = \frac{- 32 \pm i \cdot \sqrt{32^{2} \cdot 3}}{2 \cdot 1024}$

Step 4.6.2.3.1.8

Pull terms out from under the radical.

$x = \frac{- 32 \pm i \cdot (32 \sqrt{3})}{2 \cdot 1024}$

Step 4.6.2.3.1.9

Move $32$ to the left of $i$ .

$x = \frac{- 32 \pm 32 i \sqrt{3}}{2 \cdot 1024}$

Step 4.6.2.3.2

Multiply $2$ by $1024$ .

$x = \frac{- 32 \pm 32 i \sqrt{3}}{2048}$

Step 4.6.2.3.3

Simplify $\frac{- 32 \pm 32 i \sqrt{3}}{2048}$ .

$x = \frac{- 1 \pm i \sqrt{3}}{64}$

Step 4.6.2.4

The final answer is the combination of both solutions.

$x = - \frac{1 - i \sqrt{3}}{64}, - \frac{1 + i \sqrt{3}}{64}$

Step 4.7

The final solution is all the values that make $(32 x - 1) (1024 x^{2} + 32 x + 1) = 0$ true.

$x = \frac{1}{32}, - \frac{1 - i \sqrt{3}}{64}, - \frac{1 + i \sqrt{3}}{64}$