c.) $$ \frac{9^{x}-9}{3^{x}-3} $$

Question

UpStudy AI · Accepted Answer

To simplify the expression:

$$
\frac{9^{x} - 9}{3^{x} - 3}
$$

**Step 1: Rewrite in terms of powers of 3**

Since $$9 = 3^2$$, we can rewrite the expression:

$$
\frac{(3^2)^{x} - 3^2}{3^{x} - 3} = \frac{3^{2x} - 3^2}{3^{x} - 3}
$$

**Step 2: Factor the numerator**

Notice that the numerator is a difference of squares:

$$
3^{2x} - 3^2 = (3^{x})^2 - 3^2 = (3^{x} - 3)(3^{x} + 3)
$$

**Step 3: Simplify the expression**

Now, substitute the factored form back into the original expression:

$$
\frac{(3^{x} - 3)(3^{x} + 3)}{3^{x} - 3}
$$

Cancel out the common factor $$(3^{x} - 3)$$ in the numerator and the denominator:

$$
3^{x} + 3 \quad \text{(for } 3^{x} \neq 3 \text{, i.e., } x \neq 1\text{)}
$$

**Final Simplified Form:**

$$
3^{x} + 3
$$

**Answer:** After simplifying, the expression is equal to 3ˣ plus three. Thus,

3^x + 3
The simplified form of the expression is $$3^{x} + 3$$.

c.) \( \frac{9^{x}-9}{3^{x}-3} \)