Algebra Examples

$9^{2 x} \cdot 27^{3 - x} = \frac{1}{9}$

Step 1

Raise $9$ to the power of $1$ .

$9^{2 x} \cdot 27^{3 - x} = \frac{1}{9^{1}}$

Step 2

Move $9^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$9^{2 x} \cdot 27^{3 - x} = 9^{- 1}$

Step 3

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

${(3^{2})}^{2 x} \cdot 27^{3 - x} = 9^{- 1}$

Step 4

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

Tap for more steps...

Step 4.1

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

$3^{2 (2 x)} \cdot 27^{3 - x} = 9^{- 1}$

Step 4.2

Multiply $2$ by $2$ .

$3^{4 x} \cdot 27^{3 - x} = 9^{- 1}$

Step 5

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

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

Step 6

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

Tap for more steps...

Step 6.1

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

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Multiply $3$ by $3$ .

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

Step 6.4

Multiply $- 1$ by $3$ .

$3^{4 x} \cdot 3^{9 - 3 x} = 9^{- 1}$

Step 7

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

$3^{4 x + 9 - 3 x} = 9^{- 1}$

Step 8

Subtract $3 x$ from $4 x$ .

$3^{x + 9} = 9^{- 1}$

Step 9

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

$3^{x + 9} = 3^{- 2}$

Step 10

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

$x + 9 = - 2$

Step 11

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

Tap for more steps...

Step 11.1

Subtract $9$ from both sides of the equation.

$x = - 2 - 9$

Step 11.2

Subtract $9$ from $- 2$ .

$x = - 11$