Simplify 3a2 divided by a.

Step-by-step solution

1. 1

   Split the quotient into a number part and a variable part. $\frac{3a^2}{a} = 3 \cdot \frac{a^2}{a}.$ The coefficient $3$ has no $a$ in it, so it is simply carried along.

2. 2

   Apply the quotient rule for exponents. $\dfrac{a^m}{a^n} = a^{m-n}$, so with $m=2$ and $n=1$: $\frac{a^2}{a} = a^{2-1} = a^1 = a.$

3. 3

   See why the rule works, by cancelling. $a^2 = a \cdot a$, so $\dfrac{3a\cdot a}{a}$ has one factor of $a$ in both numerator and denominator; removing the matching pair leaves $3a$. Cancelling is only legitimate when that factor is nonzero.

4. 4

   Write the simplified expression. $\frac{3a^2}{a} = 3a.$

5. 5

   State the domain restriction. The original expression has $a$ in the denominator, so it is undefined at $a=0$; the simplified form $3a$ is defined there. The two agree only for $a \neq 0$, which must be recorded alongside the answer.

6. 6

   Test with a value. At $a=4$: $\dfrac{3\cdot16}{4} = \dfrac{48}{4} = 12$, and $3\cdot4 = 12$. At $a=-2$: $\dfrac{3\cdot4}{-2} = -6$, and $3(-2) = -6$ - the simplification holds for negatives too.