Precalculus Examples

$\frac{9^{4 x - 1}}{27^{x + 3}} = 81$

Step 1

Move $27^{x + 3}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$9^{4 x - 1} \cdot 27^{- (x + 3)} = 81$

Step 2

Rewrite $9$ as $3^{2}$ .

${(3^{2})}^{4 x - 1} \cdot 27^{- (x + 3)} = 81$

Step 3

Multiply the exponents in ${(3^{2})}^{4 x - 1}$ .

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Step 3.1

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

$3^{2 (4 x - 1)} \cdot 27^{- (x + 3)} = 81$

Step 3.2

Apply the distributive property.

$3^{2 (4 x) + 2 \cdot - 1} \cdot 27^{- (x + 3)} = 81$

Step 3.3

Multiply $4$ by $2$ .

$3^{8 x + 2 \cdot - 1} \cdot 27^{- (x + 3)} = 81$

Step 3.4

Multiply $2$ by $- 1$ .

$3^{8 x - 2} \cdot 27^{- (x + 3)} = 81$

Step 4

Rewrite $27$ as $3^{3}$ .

$3^{8 x - 2} \cdot {(3^{3})}^{- (x + 3)} = 81$

Step 5

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

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Step 5.1

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

$3^{8 x - 2} \cdot 3^{3 (- (x + 3))} = 81$

Step 5.2

Apply the distributive property.

$3^{8 x - 2} \cdot 3^{3 (- x - 1 \cdot 3)} = 81$

Step 5.3

Multiply $- 1$ by $3$ .

$3^{8 x - 2} \cdot 3^{3 (- x - 3)} = 81$

Step 5.4

Apply the distributive property.

$3^{8 x - 2} \cdot 3^{3 (- x) + 3 \cdot - 3} = 81$

Step 5.5

Multiply $- 1$ by $3$ .

$3^{8 x - 2} \cdot 3^{- 3 x + 3 \cdot - 3} = 81$

Step 5.6

Multiply $3$ by $- 3$ .

$3^{8 x - 2} \cdot 3^{- 3 x - 9} = 81$

Step 6

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

$3^{8 x - 2 - 3 x - 9} = 81$

Step 7

Subtract $3 x$ from $8 x$ .

$3^{5 x - 2 - 9} = 81$

Step 8

Subtract $9$ from $- 2$ .

$3^{5 x - 11} = 81$

Step 9

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

$3^{5 x - 11} = 3^{4}$

Step 10

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

$5 x - 11 = 4$

Step 11

Solve for $x$ .

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Step 11.1

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

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Step 11.1.1

Add $11$ to both sides of the equation.

$5 x = 4 + 11$

Step 11.1.2

Add $4$ and $11$ .

$5 x = 15$

Step 11.2

Divide each term in $5 x = 15$ by $5$ and simplify.

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Step 11.2.1

Divide each term in $5 x = 15$ by $5$ .

$\frac{5 x}{5} = \frac{15}{5}$

Step 11.2.2

Simplify the left side.

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Step 11.2.2.1

Cancel the common factor of $5$ .

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Step 11.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{15}{5}$

Step 11.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{15}{5}$

Step 11.2.3

Simplify the right side.

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Step 11.2.3.1

Divide $15$ by $5$ .

$x = 3$