Finite Math Examples

$8^{2 x - 1} = 32$

Step 1

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

$2^{3 (2 x - 1)} = 2^{5}$

Step 2

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

$3 (2 x - 1) = 5$

Step 3

Solve for $x$ .

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Step 3.1

Divide each term in $3 (2 x - 1) = 5$ by $3$ and simplify.

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Step 3.1.1

Divide each term in $3 (2 x - 1) = 5$ by $3$ .

$\frac{3 (2 x - 1)}{3} = \frac{5}{3}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $3$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{3 (2 x - 1)}{3} = \frac{5}{3}$

Step 3.1.2.1.2

Divide $2 x - 1$ by $1$ .

$2 x - 1 = \frac{5}{3}$

Step 3.2

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

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Step 3.2.1

Add $1$ to both sides of the equation.

$2 x = \frac{5}{3} + 1$

Step 3.2.2

Write $1$ as a fraction with a common denominator.

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

Step 3.2.3

Combine the numerators over the common denominator.

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

Step 3.2.4

Add $5$ and $3$ .

$2 x = \frac{8}{3}$

Step 3.3

Divide each term in $2 x = \frac{8}{3}$ by $2$ and simplify.

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Step 3.3.1

Divide each term in $2 x = \frac{8}{3}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{8}{3}}{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $2$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{8}{3}}{2}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{8}{3}}{2}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{8}{3} \cdot \frac{1}{2}$

Step 3.3.3.2

Cancel the common factor of $2$ .

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Step 3.3.3.2.1

Factor $2$ out of $8$ .

$x = \frac{2 (4)}{3} \cdot \frac{1}{2}$

Step 3.3.3.2.2

Cancel the common factor.

$x = \frac{2 \cdot 4}{3} \cdot \frac{1}{2}$

Step 3.3.3.2.3

Rewrite the expression.

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 1.\overline{3}$

Mixed Number Form:

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