Simplify (a^2-6)/(15a^3b^2)+3/(10ab^2)

$\frac{a^{2} - 6}{15 a^{3} b^{2}} + \frac{3}{10 a b^{2}}$

Step 1

To write $\frac{a^{2} - 6}{15 a^{3} b^{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{a^{2} - 6}{15 a^{3} b^{2}} \cdot \frac{2}{2} + \frac{3}{10 a b^{2}}$

Step 2

To write $\frac{3}{10 a b^{2}}$ as a fraction with a common denominator, multiply by $\frac{3 a^{2}}{3 a^{2}}$ .

$\frac{a^{2} - 6}{15 a^{3} b^{2}} \cdot \frac{2}{2} + \frac{3}{10 a b^{2}} \cdot \frac{3 a^{2}}{3 a^{2}}$

Step 3

Write each expression with a common denominator of $30 a^{3} b^{2}$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{a^{2} - 6}{15 a^{3} b^{2}}$ by $\frac{2}{2}$ .

$\frac{(a^{2} - 6) \cdot 2}{15 a^{3} b^{2} \cdot 2} + \frac{3}{10 a b^{2}} \cdot \frac{3 a^{2}}{3 a^{2}}$

Multiply $2$ by $15$ .

$\frac{(a^{2} - 6) \cdot 2}{30 a^{3} b^{2}} + \frac{3}{10 a b^{2}} \cdot \frac{3 a^{2}}{3 a^{2}}$

Multiply $\frac{3}{10 a b^{2}}$ by $\frac{3 a^{2}}{3 a^{2}}$ .

$\frac{(a^{2} - 6) \cdot 2}{30 a^{3} b^{2}} + \frac{3 (3 a^{2})}{10 a b^{2} (3 a^{2})}$

Multiply $3$ by $10$ .

$\frac{(a^{2} - 6) \cdot 2}{30 a^{3} b^{2}} + \frac{3 (3 a^{2})}{30 a b^{2} a^{2}}$

Multiply $a$ by $a^{2}$ by adding the exponents.

Move $a^{2}$ .

$\frac{(a^{2} - 6) \cdot 2}{30 a^{3} b^{2}} + \frac{3 (3 a^{2})}{30 (a^{2} a) b^{2}}$

Multiply $a^{2}$ by $a$ .

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

$\frac{(a^{2} - 6) \cdot 2}{30 a^{3} b^{2}} + \frac{3 (3 a^{2})}{30 (a^{2} a^{1}) b^{2}}$

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

$\frac{(a^{2} - 6) \cdot 2}{30 a^{3} b^{2}} + \frac{3 (3 a^{2})}{30 a^{2 + 1} b^{2}}$

Add $2$ and $1$ .

$\frac{(a^{2} - 6) \cdot 2}{30 a^{3} b^{2}} + \frac{3 (3 a^{2})}{30 a^{3} b^{2}}$

Step 4

Combine the numerators over the common denominator.

$\frac{(a^{2} - 6) \cdot 2 + 3 (3 a^{2})}{30 a^{3} b^{2}}$

Step 5

Simplify the numerator.

Apply the distributive property.

$\frac{a^{2} \cdot 2 - 6 \cdot 2 + 3 \cdot 3 a^{2}}{30 a^{3} b^{2}}$

Move $2$ to the left of $a^{2}$ .

$\frac{2 \cdot a^{2} - 6 \cdot 2 + 3 \cdot 3 a^{2}}{30 a^{3} b^{2}}$

Multiply $- 6$ by $2$ .

$\frac{2 \cdot a^{2} - 12 + 3 \cdot 3 a^{2}}{30 a^{3} b^{2}}$

Multiply $3$ by $3$ .

$\frac{2 a^{2} - 12 + 9 a^{2}}{30 a^{3} b^{2}}$

Add $2 a^{2}$ and $9 a^{2}$ .

$\frac{11 a^{2} - 12}{30 a^{3} b^{2}}$