Simplify (a+1)/(2a)+3/(a^2)

$\frac{a + 1}{2 a} + \frac{3}{a^{2}}$

Step 1

To write $\frac{a + 1}{2 a}$ as a fraction with a common denominator, multiply by $\frac{a}{a}$ .

$\frac{a + 1}{2 a} \cdot \frac{a}{a} + \frac{3}{a^{2}}$

Step 2

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

$\frac{a + 1}{2 a} \cdot \frac{a}{a} + \frac{3}{a^{2}} \cdot \frac{2}{2}$

Step 3

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

Multiply $\frac{a + 1}{2 a}$ by $\frac{a}{a}$ .

$\frac{(a + 1) a}{2 a \cdot a} + \frac{3}{a^{2}} \cdot \frac{2}{2}$

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

$\frac{(a + 1) a}{2 (a^{1} a)} + \frac{3}{a^{2}} \cdot \frac{2}{2}$

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

$\frac{(a + 1) a}{2 (a^{1} a^{1})} + \frac{3}{a^{2}} \cdot \frac{2}{2}$

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

$\frac{(a + 1) a}{2 a^{1 + 1}} + \frac{3}{a^{2}} \cdot \frac{2}{2}$

Add $1$ and $1$ .

$\frac{(a + 1) a}{2 a^{2}} + \frac{3}{a^{2}} \cdot \frac{2}{2}$

Multiply $\frac{3}{a^{2}}$ by $\frac{2}{2}$ .

$\frac{(a + 1) a}{2 a^{2}} + \frac{3 \cdot 2}{a^{2} \cdot 2}$

Reorder the factors of $a^{2} \cdot 2$ .

$\frac{(a + 1) a}{2 a^{2}} + \frac{3 \cdot 2}{2 a^{2}}$

Step 4

Combine the numerators over the common denominator.

$\frac{(a + 1) a + 3 \cdot 2}{2 a^{2}}$

Step 5

Simplify the numerator.

Apply the distributive property.

$\frac{a \cdot a + 1 a + 3 \cdot 2}{2 a^{2}}$

Multiply $a$ by $a$ .

$\frac{a^{2} + 1 a + 3 \cdot 2}{2 a^{2}}$

Multiply $a$ by $1$ .

$\frac{a^{2} + a + 3 \cdot 2}{2 a^{2}}$

Multiply $3$ by $2$ .

$\frac{a^{2} + a + 6}{2 a^{2}}$