Give step-by-step solution with explanation and final answer:
(5)^4 . (5)^? = (5)^9

Question

VIPSolutions · Accepted Answer

# Step-by-Step Solution

Let's solve the equation:
$$
(5)^4 \cdot (5)^{?} = (5)^9
$$

## Step 1: Use the Law of Exponents

Recall the exponent rule:
$$
a^m \cdot a^n = a^{m+n}
$$

Apply this to the left side:
$$
(5)^4 \cdot (5)^{?} = 5^{4 + ?}
$$

## Step 2: Set the Exponents Equal

Since the bases are the same and the expressions are equal,
$$
5^{4 + ?} = 5^9
$$
So, the exponents must be equal:
$$
4 + ? = 9
$$

## Step 3: Solve for the Unknown

Subtract 4 from both sides:
$$
? = 9 - 4
$$
$$
? = 5
$$

---

## Final Answer

$$
\boxed{5}
$$

**Explanation:**  
The missing exponent is **5** because $5^4 \cdot 5^5 = 5^{4+5} = 5^9$.

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