Algebra Examples

$4^{5 x - 13} = \frac{1}{8^{x}}$

Step 1

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

$4^{5 x - 13} = 8^{- x}$

Step 2

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

$2^{2 (5 x - 13)} = 2^{3 (- x)}$

Step 3

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

$2 (5 x - 13) = 3 (- x)$

Step 4

Solve for $x$ .

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Step 4.1

Simplify $2 (5 x - 13)$ .

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Step 4.1.1

Rewrite.

$0 + 0 + 2 (5 x - 13) = 3 (- x)$

Step 4.1.2

Simplify by adding zeros.

$2 (5 x - 13) = 3 (- x)$

Step 4.1.3

Apply the distributive property.

$2 (5 x) + 2 \cdot - 13 = 3 (- x)$

Step 4.1.4

Multiply.

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Step 4.1.4.1

Multiply $5$ by $2$ .

$10 x + 2 \cdot - 13 = 3 (- x)$

Step 4.1.4.2

Multiply $2$ by $- 13$ .

$10 x - 26 = 3 (- x)$

Step 4.2

Multiply $- 1$ by $3$ .

$10 x - 26 = - 3 x$

Step 4.3

Move all terms containing $x$ to the left side of the equation.

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Step 4.3.1

Add $3 x$ to both sides of the equation.

$10 x - 26 + 3 x = 0$

Step 4.3.2

Add $10 x$ and $3 x$ .

$13 x - 26 = 0$

Step 4.4

Add $26$ to both sides of the equation.

$13 x = 26$

Step 4.5

Divide each term in $13 x = 26$ by $13$ and simplify.

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Step 4.5.1

Divide each term in $13 x = 26$ by $13$ .

$\frac{13 x}{13} = \frac{26}{13}$

Step 4.5.2

Simplify the left side.

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Step 4.5.2.1

Cancel the common factor of $13$ .

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Step 4.5.2.1.1

Cancel the common factor.

$\frac{13 x}{13} = \frac{26}{13}$

Step 4.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{26}{13}$

Step 4.5.3

Simplify the right side.

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Step 4.5.3.1

Divide $26$ by $13$ .

$x = 2$