Algebra Examples

${(\frac{2}{3})}^{x + 4} - {(\frac{2}{3})}^{x}$

Step 1

Simplify each term.

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Step 1.1

Apply the product rule to $\frac{2}{3}$ .

$\frac{2^{x + 4}}{3^{x + 4}} - {(\frac{2}{3})}^{x}$

Step 1.2

Apply the product rule to $\frac{2}{3}$ .

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

Step 2

To write $\frac{2^{x + 4}}{3^{x + 4}}$ as a fraction with a common denominator, multiply by $\frac{3^{x}}{3^{x}}$ .

$\frac{2^{x + 4}}{3^{x + 4}} \cdot \frac{3^{x}}{3^{x}} - \frac{2^{x}}{3^{x}}$

Step 3

To write $- \frac{2^{x}}{3^{x}}$ as a fraction with a common denominator, multiply by $\frac{3^{4 + x}}{3^{4 + x}}$ .

$\frac{2^{x + 4}}{3^{x + 4}} \cdot \frac{3^{x}}{3^{x}} - \frac{2^{x}}{3^{x}} \cdot \frac{3^{4 + x}}{3^{4 + x}}$

Step 4

Write each expression with a common denominator of $3^{x + 4} \cdot 3^{x}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{2^{x + 4}}{3^{x + 4}}$ by $\frac{3^{x}}{3^{x}}$ .

$\frac{2^{x + 4} \cdot 3^{x}}{3^{x + 4} \cdot 3^{x}} - \frac{2^{x}}{3^{x}} \cdot \frac{3^{4 + x}}{3^{4 + x}}$

Step 4.2

Multiply $3^{x + 4}$ by $3^{x}$ by adding the exponents.

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Step 4.2.1

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

$\frac{2^{x + 4} \cdot 3^{x}}{3^{x + 4 + x}} - \frac{2^{x}}{3^{x}} \cdot \frac{3^{4 + x}}{3^{4 + x}}$

Step 4.2.2

Add $x$ and $x$ .

$\frac{2^{x + 4} \cdot 3^{x}}{3^{2 x + 4}} - \frac{2^{x}}{3^{x}} \cdot \frac{3^{4 + x}}{3^{4 + x}}$

Step 4.3

Multiply $\frac{2^{x}}{3^{x}}$ by $\frac{3^{4 + x}}{3^{4 + x}}$ .

$\frac{2^{x + 4} \cdot 3^{x}}{3^{2 x + 4}} - \frac{2^{x} \cdot 3^{4 + x}}{3^{x} \cdot 3^{4 + x}}$

Step 4.4

Multiply $3^{x}$ by $3^{4 + x}$ by adding the exponents.

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Step 4.4.1

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

$\frac{2^{x + 4} \cdot 3^{x}}{3^{2 x + 4}} - \frac{2^{x} \cdot 3^{4 + x}}{3^{x + 4 + x}}$

Step 4.4.2

Add $x$ and $x$ .

$\frac{2^{x + 4} \cdot 3^{x}}{3^{2 x + 4}} - \frac{2^{x} \cdot 3^{4 + x}}{3^{2 x + 4}}$

Step 5

Combine the numerators over the common denominator.

$\frac{2^{x + 4} \cdot 3^{x} - 2^{x} \cdot 3^{4 + x}}{3^{2 x + 4}}$