Algebra Examples

${(\frac{4}{9})}^{x} {(\frac{27}{8})}^{x - 1} = (\frac{2}{3})$

Step 1

Rewrite $\frac{4}{9}$ as ${(\frac{2}{3})}^{2}$ .

${({(\frac{2}{3})}^{2})}^{x} {(\frac{27}{8})}^{x - 1} = \frac{2}{3}$

Step 2

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

${(\frac{2}{3})}^{2 x} {(\frac{27}{8})}^{x - 1} = \frac{2}{3}$

Step 3

Rewrite $\frac{27}{8}$ as ${(\frac{8}{27})}^{- 1}$ .

${(\frac{2}{3})}^{2 x} {({(\frac{8}{27})}^{- 1})}^{x - 1} = \frac{2}{3}$

Step 4

Rewrite $\frac{8}{27}$ as ${(\frac{2}{3})}^{3}$ .

${(\frac{2}{3})}^{2 x} {({({(\frac{2}{3})}^{3})}^{- 1})}^{x - 1} = \frac{2}{3}$

Step 5

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

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

Step 6

Multiply $3$ by $- 1$ .

${(\frac{2}{3})}^{2 x} {({(\frac{2}{3})}^{- 3})}^{x - 1} = \frac{2}{3}$

Step 7

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

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Step 7.1

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

${(\frac{2}{3})}^{2 x} {(\frac{2}{3})}^{- 3 (x - 1)} = \frac{2}{3}$

Step 7.2

Apply the distributive property.

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

Step 7.3

Multiply $- 3$ by $- 1$ .

${(\frac{2}{3})}^{2 x} {(\frac{2}{3})}^{- 3 x + 3} = \frac{2}{3}$

Step 8

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

${(\frac{2}{3})}^{2 x - 3 x + 3} = \frac{2}{3}$

Step 9

Subtract $3 x$ from $2 x$ .

${(\frac{2}{3})}^{- x + 3} = \frac{2}{3}$

Step 10

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

${(\frac{2}{3})}^{- x + 3} = {(\frac{2}{3})}^{1}$

Step 11

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

$- x + 3 = 1$

Step 12

Solve for $x$ .

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Step 12.1

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

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Step 12.1.1

Subtract $3$ from both sides of the equation.

$- x = 1 - 3$

Step 12.1.2

Subtract $3$ from $1$ .

$- x = - 2$

Step 12.2

Divide each term in $- x = - 2$ by $- 1$ and simplify.

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Step 12.2.1

Divide each term in $- x = - 2$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 2}{- 1}$

Step 12.2.2

Simplify the left side.

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Step 12.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 2}{- 1}$

Step 12.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 2}{- 1}$

Step 12.2.3

Simplify the right side.

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Step 12.2.3.1

Divide $- 2$ by $- 1$ .

$x = 2$