Algebra Examples

$8^{5 x + 3} = {(\frac{1}{64})}^{- \frac{2}{3}}$

Step 1

Apply the product rule to $\frac{1}{64}$ .

$8^{5 x + 3} = \frac{1^{- \frac{2}{3}}}{64^{- \frac{2}{3}}}$

Step 2

One to any power is one.

$8^{5 x + 3} = \frac{1}{64^{- \frac{2}{3}}}$

Step 3

Move $64^{- \frac{2}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$8^{5 x + 3} = 64^{\frac{2}{3}}$

Step 4

Create equivalent expressions in the equation that all have equal bases.

$8^{5 x + 3} = 8^{2 (\frac{2}{3})}$

Step 5

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$5 x + 3 = 2 (\frac{2}{3})$

Step 6

Solve for $x$ .

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Step 6.1

Multiply $2 (\frac{2}{3})$ .

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Step 6.1.1

Combine $2$ and $\frac{2}{3}$ .

$5 x + 3 = \frac{2 \cdot 2}{3}$

Step 6.1.2

Multiply $2$ by $2$ .

$5 x + 3 = \frac{4}{3}$

Step 6.2

Move all terms not containing $x$ to the right side of the equation.

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Step 6.2.1

Subtract $3$ from both sides of the equation.

$5 x = \frac{4}{3} - 3$

Step 6.2.2

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

$5 x = \frac{4}{3} - 3 \cdot \frac{3}{3}$

Step 6.2.3

Combine $- 3$ and $\frac{3}{3}$ .

$5 x = \frac{4}{3} + \frac{- 3 \cdot 3}{3}$

Step 6.2.4

Combine the numerators over the common denominator.

$5 x = \frac{4 - 3 \cdot 3}{3}$

Step 6.2.5

Simplify the numerator.

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Step 6.2.5.1

Multiply $- 3$ by $3$ .

$5 x = \frac{4 - 9}{3}$

Step 6.2.5.2

Subtract $9$ from $4$ .

$5 x = \frac{- 5}{3}$

Step 6.2.6

Move the negative in front of the fraction.

$5 x = - \frac{5}{3}$

Step 6.3

Divide each term in $5 x = - \frac{5}{3}$ by $5$ and simplify.

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Step 6.3.1

Divide each term in $5 x = - \frac{5}{3}$ by $5$ .

$\frac{5 x}{5} = \frac{- \frac{5}{3}}{5}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor of $5$ .

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Step 6.3.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- \frac{5}{3}}{5}$

Step 6.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{5}{3}}{5}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{5}{3} \cdot \frac{1}{5}$

Step 6.3.3.2

Cancel the common factor of $5$ .

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Step 6.3.3.2.1

Move the leading negative in $- \frac{5}{3}$ into the numerator.

$x = \frac{- 5}{3} \cdot \frac{1}{5}$

Step 6.3.3.2.2

Factor $5$ out of $- 5$ .

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

Step 6.3.3.2.3

Cancel the common factor.

$x = \frac{5 \cdot - 1}{3} \cdot \frac{1}{5}$

Step 6.3.3.2.4

Rewrite the expression.

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

Step 6.3.3.3

Move the negative in front of the fraction.

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 0.\overline{3}$