Question

simplify the expression (\frac{\frac{6}{a+h} - \frac{6}{a}}{h}) and give your answer in the form of (\frac{a}{b}). your answer for (a) is : (square) your answer for (b) is : (square)

Explanation:

Step1: Simplify the numerator

First, we simplify the numerator \(\frac{6}{a + h}-\frac{6}{a}\). To do this, we find a common denominator, which is \(a(a + h)\). So we have:
\[

$$\begin{align*} \frac{6}{a + h}-\frac{6}{a}&=\frac{6a}{a(a + h)}-\frac{6(a + h)}{a(a + h)}\\ &=\frac{6a-6(a + h)}{a(a + h)}\\ &=\frac{6a-6a - 6h}{a(a + h)}\\ &=\frac{-6h}{a(a + h)} \end{align*}$$

\]

Step2: Divide by \(h\)

Now our original expression \(\frac{\frac{6}{a + h}-\frac{6}{a}}{h}\) becomes \(\frac{\frac{-6h}{a(a + h)}}{h}\). Dividing by \(h\) is the same as multiplying by \(\frac{1}{h}\), so:
\[
\frac{\frac{-6h}{a(a + h)}}{h}=\frac{-6h}{a(a + h)}\times\frac{1}{h}=\frac{-6}{a(a + h)}
\]

Answer:

For \(A\): \(- 6\)
For \(B\): \(a(a + h)\) (or \(a^{2}+ah\))