Algebra Examples

$\frac{1}{16} = 64^{4 x - 3}$

Step 1

Rewrite the equation as $64^{4 x - 3} = \frac{1}{16}$ .

$64^{4 x - 3} = \frac{1}{16}$

Step 2

Raise $16$ to the power of $1$ .

$64^{4 x - 3} = \frac{1}{16^{1}}$

Step 3

Move $16^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$64^{4 x - 3} = 16^{- 1}$

Step 4

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

$4^{3 (4 x - 3)} = 4^{- 2}$

Step 5

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

$3 (4 x - 3) = - 2$

Step 6

Solve for $x$ .

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Step 6.1

Divide each term in $3 (4 x - 3) = - 2$ by $3$ and simplify.

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Step 6.1.1

Divide each term in $3 (4 x - 3) = - 2$ by $3$ .

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

Step 6.1.2

Simplify the left side.

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Step 6.1.2.1

Cancel the common factor of $3$ .

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Step 6.1.2.1.1

Cancel the common factor.

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

Step 6.1.2.1.2

Divide $4 x - 3$ by $1$ .

$4 x - 3 = \frac{- 2}{3}$

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.

$4 x - 3 = - \frac{2}{3}$

Step 6.2

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

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Step 6.2.1

Add $3$ to both sides of the equation.

$4 x = - \frac{2}{3} + 3$

Step 6.2.2

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

$4 x = - \frac{2}{3} + 3 \cdot \frac{3}{3}$

Step 6.2.3

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

$4 x = - \frac{2}{3} + \frac{3 \cdot 3}{3}$

Step 6.2.4

Combine the numerators over the common denominator.

$4 x = \frac{- 2 + 3 \cdot 3}{3}$

Step 6.2.5

Simplify the numerator.

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Step 6.2.5.1

Multiply $3$ by $3$ .

$4 x = \frac{- 2 + 9}{3}$

Step 6.2.5.2

Add $- 2$ and $9$ .

$4 x = \frac{7}{3}$

Step 6.3

Divide each term in $4 x = \frac{7}{3}$ by $4$ and simplify.

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Step 6.3.1

Divide each term in $4 x = \frac{7}{3}$ by $4$ .

$\frac{4 x}{4} = \frac{\frac{7}{3}}{4}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor of $4$ .

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Step 6.3.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{\frac{7}{3}}{4}$

Step 6.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{7}{3}}{4}$

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.

$x = \frac{7}{3} \cdot \frac{1}{4}$

Step 6.3.3.2

Multiply $\frac{7}{3} \cdot \frac{1}{4}$ .

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Step 6.3.3.2.1

Multiply $\frac{7}{3}$ by $\frac{1}{4}$ .

$x = \frac{7}{3 \cdot 4}$

Step 6.3.3.2.2

Multiply $3$ by $4$ .

$x = \frac{7}{12}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{7}{12}$

Decimal Form:

$x = 0.58\overline{3}$