Pre-Algebra Examples

$8 x^{\frac{1}{3}} = x^{- \frac{2}{3}}$

Step 1

Eliminate the fractional exponents by multiplying both exponents by the LCD.

${(8 x^{\frac{1}{3}})}^{3} = {(x^{- \frac{2}{3}})}^{3}$

Step 2

Simplify ${(8 x^{\frac{1}{3}})}^{3}$ .

Tap for more steps...

Step 2.1

Apply the product rule to $8 x^{\frac{1}{3}}$ .

$8^{3} {(x^{\frac{1}{3}})}^{3} = {(x^{- \frac{2}{3}})}^{3}$

Step 2.2

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

$512 {(x^{\frac{1}{3}})}^{3} = {(x^{- \frac{2}{3}})}^{3}$

Step 2.3

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

Tap for more steps...

Step 2.3.1

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

$512 x^{\frac{1}{3} \cdot 3} = {(x^{- \frac{2}{3}})}^{3}$

Step 2.3.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.3.2.1

Cancel the common factor.

$512 x^{\frac{1}{3} \cdot 3} = {(x^{- \frac{2}{3}})}^{3}$

Step 2.3.2.2

Rewrite the expression.

$512 x^{1} = {(x^{- \frac{2}{3}})}^{3}$

Step 2.4

Simplify.

$512 x = {(x^{- \frac{2}{3}})}^{3}$

Step 3

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

Tap for more steps...

Step 3.1

Multiply the exponents in ${(x^{- \frac{2}{3}})}^{3}$ .

Tap for more steps...

Step 3.1.1

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

$512 x = x^{- \frac{2}{3} \cdot 3}$

Step 3.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.1.2.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$512 x = x^{\frac{- 2}{3} \cdot 3}$

Step 3.1.2.2

Cancel the common factor.

$512 x = x^{\frac{- 2}{3} \cdot 3}$

Step 3.1.2.3

Rewrite the expression.

$512 x = x^{- 2}$

Step 3.2

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

$512 x = \frac{1}{x^{2}}$

Step 4

Find the LCD of the terms in the equation.

Tap for more steps...

Step 4.1

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

$1, x^{2}$

Step 4.2

The LCM of one and any expression is the expression.

$x^{2}$

Step 5

Multiply each term in $512 x = \frac{1}{x^{2}}$ by $x^{2}$ to eliminate the fractions.

Tap for more steps...

Step 5.1

Multiply each term in $512 x = \frac{1}{x^{2}}$ by $x^{2}$ .

$512 x \cdot x^{2} = \frac{1}{x^{2}} x^{2}$

Step 5.2

Simplify the left side.

Tap for more steps...

Step 5.2.1

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

Tap for more steps...

Step 5.2.1.1

Move $x^{2}$ .

$512 (x^{2} x) = \frac{1}{x^{2}} x^{2}$

Step 5.2.1.2

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

Tap for more steps...

Step 5.2.1.2.1

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

$512 (x^{2} x^{1}) = \frac{1}{x^{2}} x^{2}$

Step 5.2.1.2.2

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

$512 x^{2 + 1} = \frac{1}{x^{2}} x^{2}$

Step 5.2.1.3

Add $2$ and $1$ .

$512 x^{3} = \frac{1}{x^{2}} x^{2}$

Step 5.3

Simplify the right side.

Tap for more steps...

Step 5.3.1

Cancel the common factor of $x^{2}$ .

Tap for more steps...

Step 5.3.1.1

Cancel the common factor.

$512 x^{3} = \frac{1}{x^{2}} x^{2}$

Step 5.3.1.2

Rewrite the expression.

$512 x^{3} = 1$

Step 6

Solve the equation.

Tap for more steps...

Step 6.1

Subtract $1$ from both sides of the equation.

$512 x^{3} - 1 = 0$

Step 6.2

Factor the left side of the equation.

Tap for more steps...

Step 6.2.1

Rewrite $512 x^{3}$ as ${(8 x)}^{3}$ .

${(8 x)}^{3} - 1 = 0$

Step 6.2.2

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

${(8 x)}^{3} - 1^{3} = 0$

Step 6.2.3

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 $a = 8 x$ and $b = 1$ .

$(8 x - 1) ({(8 x)}^{2} + 8 x \cdot 1 + 1^{2}) = 0$

Step 6.2.4

Simplify.

Tap for more steps...

Step 6.2.4.1

Apply the product rule to $8 x$ .

$(8 x - 1) (8^{2} x^{2} + 8 x \cdot 1 + 1^{2}) = 0$

Step 6.2.4.2

Raise $8$ to the power of $2$ .

$(8 x - 1) (64 x^{2} + 8 x \cdot 1 + 1^{2}) = 0$

Step 6.2.4.3

Multiply $8$ by $1$ .

$(8 x - 1) (64 x^{2} + 8 x + 1^{2}) = 0$

Step 6.2.4.4

One to any power is one.

$(8 x - 1) (64 x^{2} + 8 x + 1) = 0$

Step 6.3

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

$8 x - 1 = 0$

$64 x^{2} + 8 x + 1 = 0$

Step 6.4

Set $8 x - 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 6.4.1

Set $8 x - 1$ equal to $0$ .

$8 x - 1 = 0$

Step 6.4.2

Solve $8 x - 1 = 0$ for $x$ .

Tap for more steps...

Step 6.4.2.1

Add $1$ to both sides of the equation.

$8 x = 1$

Step 6.4.2.2

Divide each term in $8 x = 1$ by $8$ and simplify.

Tap for more steps...

Step 6.4.2.2.1

Divide each term in $8 x = 1$ by $8$ .

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

Step 6.4.2.2.2

Simplify the left side.

Tap for more steps...

Step 6.4.2.2.2.1

Cancel the common factor of $8$ .

Tap for more steps...

Step 6.4.2.2.2.1.1

Cancel the common factor.

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

Step 6.4.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 6.5

Set $64 x^{2} + 8 x + 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 6.5.1

Set $64 x^{2} + 8 x + 1$ equal to $0$ .

$64 x^{2} + 8 x + 1 = 0$

Step 6.5.2

Solve $64 x^{2} + 8 x + 1 = 0$ for $x$ .

Tap for more steps...

Step 6.5.2.1

Use the quadratic formula to find the solutions.

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

Step 6.5.2.2

Substitute the values $a = 64$ , $b = 8$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{- 8 \pm \sqrt{8^{2} - 4 \cdot (64 \cdot 1)}}{2 \cdot 64}$

Step 6.5.2.3

Simplify.

Tap for more steps...

Step 6.5.2.3.1

Simplify the numerator.

Tap for more steps...

Step 6.5.2.3.1.1

Raise $8$ to the power of $2$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 64 \cdot 1}}{2 \cdot 64}$

Step 6.5.2.3.1.2

Multiply $- 4 \cdot 64 \cdot 1$ .

Tap for more steps...

Step 6.5.2.3.1.2.1

Multiply $- 4$ by $64$ .

$x = \frac{- 8 \pm \sqrt{64 - 256 \cdot 1}}{2 \cdot 64}$

Step 6.5.2.3.1.2.2

Multiply $- 256$ by $1$ .

$x = \frac{- 8 \pm \sqrt{64 - 256}}{2 \cdot 64}$

Step 6.5.2.3.1.3

Subtract $256$ from $64$ .

$x = \frac{- 8 \pm \sqrt{- 192}}{2 \cdot 64}$

Step 6.5.2.3.1.4

Rewrite $- 192$ as $- 1 (192)$ .

$x = \frac{- 8 \pm \sqrt{- 1 \cdot 192}}{2 \cdot 64}$

Step 6.5.2.3.1.5

Rewrite $\sqrt{- 1 (192)}$ as $\sqrt{- 1} \cdot \sqrt{192}$ .

$x = \frac{- 8 \pm \sqrt{- 1} \cdot \sqrt{192}}{2 \cdot 64}$

Step 6.5.2.3.1.6

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

$x = \frac{- 8 \pm i \cdot \sqrt{192}}{2 \cdot 64}$

Step 6.5.2.3.1.7

Rewrite $192$ as $8^{2} \cdot 3$ .

Tap for more steps...

Step 6.5.2.3.1.7.1

Factor $64$ out of $192$ .

$x = \frac{- 8 \pm i \cdot \sqrt{64 (3)}}{2 \cdot 64}$

Step 6.5.2.3.1.7.2

Rewrite $64$ as $8^{2}$ .

$x = \frac{- 8 \pm i \cdot \sqrt{8^{2} \cdot 3}}{2 \cdot 64}$

Step 6.5.2.3.1.8

Pull terms out from under the radical.

$x = \frac{- 8 \pm i \cdot (8 \sqrt{3})}{2 \cdot 64}$

Step 6.5.2.3.1.9

Move $8$ to the left of $i$ .

$x = \frac{- 8 \pm 8 i \sqrt{3}}{2 \cdot 64}$

Step 6.5.2.3.2

Multiply $2$ by $64$ .

$x = \frac{- 8 \pm 8 i \sqrt{3}}{128}$

Step 6.5.2.3.3

Simplify $\frac{- 8 \pm 8 i \sqrt{3}}{128}$ .

$x = \frac{- 1 \pm i \sqrt{3}}{16}$

Step 6.5.2.4

The final answer is the combination of both solutions.

$x = - \frac{1 - i \sqrt{3}}{16}, - \frac{1 + i \sqrt{3}}{16}$

Step 6.6

The final solution is all the values that make $(8 x - 1) (64 x^{2} + 8 x + 1) = 0$ true.

$x = \frac{1}{8}, - \frac{1 - i \sqrt{3}}{16}, - \frac{1 + i \sqrt{3}}{16}$