Value of `(625)^{0.16} \times(625)^{0.09}` is

Options

A. 16
B. 8
C. 4
D. 5

Answer

To solve the expression \((625)^{0.16} \times (625)^{0.09}\), we can follow these steps:

### Step 1: Combine the Exponents
Using the property of exponents that states \(a^m \times a^n = a^{m+n}\), we can combine the exponents of the two terms.
\[ (625)^{0.16} \times (625)^{0.09} = (625)^{0.16 + 0.09} \]

### Step 2: Add the Exponents
Now, we add the exponents:
\[ 0.16 + 0.09 = 0.25 \]
So, we can rewrite the expression as:
\[ (625)^{0.25} \]

### Step 3: Rewrite the Base
Next, we can express \(625\) as a power of \(5\). We know that:
\[ 625 = 5^4 \]
Thus, we can substitute this into our expression:
\[ (625)^{0.25} = (5^4)^{0.25} \]

### Step 4: Apply the Power of a Power Rule
Using the property of exponents that states \((a^m)^n = a^{m \cdot n}\), we can simplify further:
\[ (5^4)^{0.25} = 5^{4 \times 0.25} = 5^{1} \]

### Step 5: Final Calculation
Now, we can compute \(5^1\):
\[ 5^1 = 5 \]

### Conclusion
Therefore, the value of \((625)^{0.16} \times (625)^{0.09}\) is:
\[ \boxed{5} \]