$$ \frac { 9 ^ { n } - 4.3 ^ { n } - 5 } { 3 ^ { n } + 1 } $$

Question

UpStudy AI · Accepted Answer

To simplify the expression:

$$
\frac{9^n - 4 \cdot 3^n - 5}{3^n + 1}
$$

**Step 1: Express $$9^n$$ in terms of $$3^n$$**

Since $$9 = 3^2$$, we can rewrite $$9^n$$ as:

$$
9^n = (3^2)^n = 3^{2n}
$$

**Step 2: Let $$x = 3^n$$**

Substituting $$x$$ for $$3^n$$, the expression becomes:

$$
\frac{x^2 - 4x - 5}{x + 1}
$$

**Step 3: Factor the numerator**

We need to factor the quadratic expression in the numerator:

$$
x^2 - 4x - 5 = (x - 5)(x + 1)
$$

**Step 4: Simplify the expression**

Substitute the factored form back into the expression:

$$
\frac{(x - 5)(x + 1)}{x + 1}
$$

We can cancel out the $$(x + 1)$$ terms (as long as $$x + 1 \neq 0$$, which is always true since $$x = 3^n > 0$$):

$$
x - 5
$$

**Step 5: Substitute back $$x = 3^n$$**

$$
3^n - 5
$$

**Final Simplified Form:**

$$
\frac{9^n - 4 \cdot 3^n - 5}{3^n + 1} = 3^n - 5
$$

---

**Example:**

If $$n = 2$$:

$$
\frac{9^2 - 4 \cdot 3^2 - 5}{3^2 + 1} = \frac{81 - 36 - 5}{9 + 1} = \frac{40}{10} = 4
$$

Using the simplified form:

$$
3^2 - 5 = 9 - 5 = 4
$$

Both methods yield the same result.
The simplified form of the expression is $$3^n - 5$$.

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