Simplify (5/2)^x+(5/2)^(x+3)

${(\frac{5}{2})}^{x} + {(\frac{5}{2})}^{x + 3}$

Step 1

Simplify each term.

Apply the product rule to $\frac{5}{2}$ .

$\frac{5^{x}}{2^{x}} + {(\frac{5}{2})}^{x + 3}$

Apply the product rule to $\frac{5}{2}$ .

$\frac{5^{x}}{2^{x}} + \frac{5^{x + 3}}{2^{x + 3}}$

Step 2

To write $\frac{5^{x}}{2^{x}}$ as a fraction with a common denominator, multiply by $\frac{2^{x + 3}}{2^{x + 3}}$ .

$\frac{5^{x}}{2^{x}} \cdot \frac{2^{x + 3}}{2^{x + 3}} + \frac{5^{x + 3}}{2^{x + 3}}$

Step 3

To write $\frac{5^{x + 3}}{2^{x + 3}}$ as a fraction with a common denominator, multiply by $\frac{2^{x}}{2^{x}}$ .

$\frac{5^{x}}{2^{x}} \cdot \frac{2^{x + 3}}{2^{x + 3}} + \frac{5^{x + 3}}{2^{x + 3}} \cdot \frac{2^{x}}{2^{x}}$

Step 4

Write each expression with a common denominator of $2^{x} \cdot 2^{x + 3}$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{5^{x}}{2^{x}}$ by $\frac{2^{x + 3}}{2^{x + 3}}$ .

$\frac{5^{x} \cdot 2^{x + 3}}{2^{x} \cdot 2^{x + 3}} + \frac{5^{x + 3}}{2^{x + 3}} \cdot \frac{2^{x}}{2^{x}}$

Multiply $2^{x}$ by $2^{x + 3}$ by adding the exponents.

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

$\frac{5^{x} \cdot 2^{x + 3}}{2^{x + x + 3}} + \frac{5^{x + 3}}{2^{x + 3}} \cdot \frac{2^{x}}{2^{x}}$

Add $x$ and $x$ .

$\frac{5^{x} \cdot 2^{x + 3}}{2^{2 x + 3}} + \frac{5^{x + 3}}{2^{x + 3}} \cdot \frac{2^{x}}{2^{x}}$

Multiply $\frac{5^{x + 3}}{2^{x + 3}}$ by $\frac{2^{x}}{2^{x}}$ .

$\frac{5^{x} \cdot 2^{x + 3}}{2^{2 x + 3}} + \frac{5^{x + 3} \cdot 2^{x}}{2^{x + 3} \cdot 2^{x}}$

Multiply $2^{x + 3}$ by $2^{x}$ by adding the exponents.

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

$\frac{5^{x} \cdot 2^{x + 3}}{2^{2 x + 3}} + \frac{5^{x + 3} \cdot 2^{x}}{2^{x + 3 + x}}$

Add $x$ and $x$ .

$\frac{5^{x} \cdot 2^{x + 3}}{2^{2 x + 3}} + \frac{5^{x + 3} \cdot 2^{x}}{2^{2 x + 3}}$

Step 5

Combine the numerators over the common denominator.

$\frac{5^{x} \cdot 2^{x + 3} + 5^{x + 3} \cdot 2^{x}}{2^{2 x + 3}}$