Finite Math Examples

$\frac{1}{8} = 4^{7 x + 5}$

Step 1

Rewrite the equation as $4^{7 x + 5} = \frac{1}{8}$ .

$4^{7 x + 5} = \frac{1}{8}$

Step 2

Raise $8$ to the power of $1$ .

$4^{7 x + 5} = \frac{1}{8^{1}}$

Step 3

Move $8^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$4^{7 x + 5} = 8^{- 1}$

Step 4

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

$2^{2 (7 x + 5)} = 2^{- 3}$

Step 5

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

$2 (7 x + 5) = - 3$

Step 6

Solve for $x$ .

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Step 6.1

Divide each term in $2 (7 x + 5) = - 3$ by $2$ and simplify.

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Step 6.1.1

Divide each term in $2 (7 x + 5) = - 3$ by $2$ .

$\frac{2 (7 x + 5)}{2} = \frac{- 3}{2}$

Step 6.1.2

Simplify the left side.

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Step 6.1.2.1

Cancel the common factor of $2$ .

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Step 6.1.2.1.1

Cancel the common factor.

$\frac{2 (7 x + 5)}{2} = \frac{- 3}{2}$

Step 6.1.2.1.2

Divide $7 x + 5$ by $1$ .

$7 x + 5 = \frac{- 3}{2}$

Step 6.1.3

Simplify the right side.

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Step 6.1.3.1

Move the negative in front of the fraction.

$7 x + 5 = - \frac{3}{2}$

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 $5$ from both sides of the equation.

$7 x = - \frac{3}{2} - 5$

Step 6.2.2

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$7 x = - \frac{3}{2} - 5 \cdot \frac{2}{2}$

Step 6.2.3

Combine $- 5$ and $\frac{2}{2}$ .

$7 x = - \frac{3}{2} + \frac{- 5 \cdot 2}{2}$

Step 6.2.4

Combine the numerators over the common denominator.

$7 x = \frac{- 3 - 5 \cdot 2}{2}$

Step 6.2.5

Simplify the numerator.

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Step 6.2.5.1

Multiply $- 5$ by $2$ .

$7 x = \frac{- 3 - 10}{2}$

Step 6.2.5.2

Subtract $10$ from $- 3$ .

$7 x = \frac{- 13}{2}$

Step 6.2.6

Move the negative in front of the fraction.

$7 x = - \frac{13}{2}$

Step 6.3

Divide each term in $7 x = - \frac{13}{2}$ by $7$ and simplify.

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Step 6.3.1

Divide each term in $7 x = - \frac{13}{2}$ by $7$ .

$\frac{7 x}{7} = \frac{- \frac{13}{2}}{7}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor of $7$ .

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Step 6.3.2.1.1

Cancel the common factor.

$\frac{7 x}{7} = \frac{- \frac{13}{2}}{7}$

Step 6.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{13}{2}}{7}$

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.

$x = - \frac{13}{2} \cdot \frac{1}{7}$

Step 6.3.3.2

Multiply $- \frac{13}{2} \cdot \frac{1}{7}$ .

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Step 6.3.3.2.1

Multiply $\frac{1}{7}$ by $\frac{13}{2}$ .

$x = - \frac{13}{7 \cdot 2}$

Step 6.3.3.2.2

Multiply $7$ by $2$ .

$x = - \frac{13}{14}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{13}{14}$

Decimal Form:

$x = - 0.9\overline{285714}$