Algebra Examples

$\frac{81^{x + 5}}{3^{7 x + 2}} = 3^{x - 2}$

Step 1

Move $3^{7 x + 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$81^{x + 5} \cdot 3^{- (7 x + 2)} = 3^{x - 2}$

Step 2

Rewrite $81$ as $3^{4}$ .

${(3^{4})}^{x + 5} \cdot 3^{- (7 x + 2)} = 3^{x - 2}$

Step 3

Multiply the exponents in ${(3^{4})}^{x + 5}$ .

Tap for more steps...

Step 3.1

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

$3^{4 (x + 5)} \cdot 3^{- (7 x + 2)} = 3^{x - 2}$

Step 3.2

Apply the distributive property.

$3^{4 x + 4 \cdot 5} \cdot 3^{- (7 x + 2)} = 3^{x - 2}$

Step 3.3

Multiply $4$ by $5$ .

$3^{4 x + 20} \cdot 3^{- (7 x + 2)} = 3^{x - 2}$

Step 4

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

$3^{4 x + 20 - (7 x + 2)} = 3^{x - 2}$

Step 5

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

$4 x + 20 - (7 x + 2) = x - 2$

Step 6

Solve for $x$ .

Tap for more steps...

Step 6.1

Simplify $4 x + 20 - (7 x + 2)$ .

Tap for more steps...

Step 6.1.1

Simplify each term.

Tap for more steps...

Step 6.1.1.1

Apply the distributive property.

$4 x + 20 - (7 x) - 1 \cdot 2 = x - 2$

Step 6.1.1.2

Multiply $7$ by $- 1$ .

$4 x + 20 - 7 x - 1 \cdot 2 = x - 2$

Step 6.1.1.3

Multiply $- 1$ by $2$ .

$4 x + 20 - 7 x - 2 = x - 2$

Step 6.1.2

Simplify by adding terms.

Tap for more steps...

Step 6.1.2.1

Subtract $7 x$ from $4 x$ .

$- 3 x + 20 - 2 = x - 2$

Step 6.1.2.2

Subtract $2$ from $20$ .

$- 3 x + 18 = x - 2$

Step 6.2

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

Tap for more steps...

Step 6.2.1

Subtract $x$ from both sides of the equation.

$- 3 x + 18 - x = - 2$

Step 6.2.2

Subtract $x$ from $- 3 x$ .

$- 4 x + 18 = - 2$

Step 6.3

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

Tap for more steps...

Step 6.3.1

Subtract $18$ from both sides of the equation.

$- 4 x = - 2 - 18$

Step 6.3.2

Subtract $18$ from $- 2$ .

$- 4 x = - 20$

Step 6.4

Divide each term in $- 4 x = - 20$ by $- 4$ and simplify.

Tap for more steps...

Step 6.4.1

Divide each term in $- 4 x = - 20$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{- 20}{- 4}$

Step 6.4.2

Simplify the left side.

Tap for more steps...

Step 6.4.2.1

Cancel the common factor of $- 4$ .

Tap for more steps...

Step 6.4.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{- 20}{- 4}$

Step 6.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 20}{- 4}$

Step 6.4.3

Simplify the right side.

Tap for more steps...

Step 6.4.3.1

Divide $- 20$ by $- 4$ .

$x = 5$