Algebra Examples

$\frac{8^{2 x + 4}}{4^{x - 3}} = 4^{x + 5}$

Step 1

Move $4^{x - 3}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$8^{2 x + 4} \cdot 4^{- (x - 3)} = 4^{x + 5}$

Step 2

Rewrite $8$ as $2^{3}$ .

${(2^{3})}^{2 x + 4} \cdot 4^{- (x - 3)} = 4^{x + 5}$

Step 3

Multiply the exponents in ${(2^{3})}^{2 x + 4}$ .

Tap for more steps...

Step 3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$2^{3 (2 x + 4)} \cdot 4^{- (x - 3)} = 4^{x + 5}$

Step 3.2

Apply the distributive property.

$2^{3 (2 x) + 3 \cdot 4} \cdot 4^{- (x - 3)} = 4^{x + 5}$

Step 3.3

Multiply $2$ by $3$ .

$2^{6 x + 3 \cdot 4} \cdot 4^{- (x - 3)} = 4^{x + 5}$

Step 3.4

Multiply $3$ by $4$ .

$2^{6 x + 12} \cdot 4^{- (x - 3)} = 4^{x + 5}$

Step 4

Rewrite $4$ as $2^{2}$ .

$2^{6 x + 12} \cdot {(2^{2})}^{- (x - 3)} = 4^{x + 5}$

Step 5

Multiply the exponents in ${(2^{2})}^{- (x - 3)}$ .

Tap for more steps...

Step 5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$2^{6 x + 12} \cdot 2^{2 (- (x - 3))} = 4^{x + 5}$

Step 5.2

Apply the distributive property.

$2^{6 x + 12} \cdot 2^{2 (- x - - 3)} = 4^{x + 5}$

Step 5.3

Multiply $- 1$ by $- 3$ .

$2^{6 x + 12} \cdot 2^{2 (- x + 3)} = 4^{x + 5}$

Step 5.4

Apply the distributive property.

$2^{6 x + 12} \cdot 2^{2 (- x) + 2 \cdot 3} = 4^{x + 5}$

Step 5.5

Multiply $- 1$ by $2$ .

$2^{6 x + 12} \cdot 2^{- 2 x + 2 \cdot 3} = 4^{x + 5}$

Step 5.6

Multiply $2$ by $3$ .

$2^{6 x + 12} \cdot 2^{- 2 x + 6} = 4^{x + 5}$

Step 6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2^{6 x + 12 - 2 x + 6} = 4^{x + 5}$

Step 7

Subtract $2 x$ from $6 x$ .

$2^{4 x + 12 + 6} = 4^{x + 5}$

Step 8

Add $12$ and $6$ .

$2^{4 x + 18} = 4^{x + 5}$

Step 9

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

$2^{4 x + 18} = 2^{2 (x + 5)}$

Step 10

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

$4 x + 18 = 2 (x + 5)$

Step 11

Solve for $x$ .

Tap for more steps...

Step 11.1

Simplify $2 (x + 5)$ .

Tap for more steps...

Step 11.1.1

Rewrite.

$4 x + 18 = 0 + 0 + 2 (x + 5)$

Step 11.1.2

Simplify by adding zeros.

$4 x + 18 = 2 (x + 5)$

Step 11.1.3

Apply the distributive property.

$4 x + 18 = 2 x + 2 \cdot 5$

Step 11.1.4

Multiply $2$ by $5$ .

$4 x + 18 = 2 x + 10$

Step 11.2

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

Tap for more steps...

Step 11.2.1

Subtract $2 x$ from both sides of the equation.

$4 x + 18 - 2 x = 10$

Step 11.2.2

Subtract $2 x$ from $4 x$ .

$2 x + 18 = 10$

Step 11.3

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

Tap for more steps...

Step 11.3.1

Subtract $18$ from both sides of the equation.

$2 x = 10 - 18$

Step 11.3.2

Subtract $18$ from $10$ .

$2 x = - 8$

Step 11.4

Divide each term in $2 x = - 8$ by $2$ and simplify.

Tap for more steps...

Step 11.4.1

Divide each term in $2 x = - 8$ by $2$ .

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

Step 11.4.2

Simplify the left side.

Tap for more steps...

Step 11.4.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 11.4.2.1.1

Cancel the common factor.

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

Step 11.4.2.1.2

Divide $x$ by $1$ .

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

Step 11.4.3

Simplify the right side.

Tap for more steps...

Step 11.4.3.1

Divide $- 8$ by $2$ .

$x = - 4$