If `(2000)^(10)=1.024xx10^(k)`, then the value of k is

Options

A. 33
B. 30
C. 34
D. 31

Answer

To solve the equation \( (2000)^{10} = 1.024 \times 10^k \), we need to find the value of \( k \).

### Step-by-Step Solution:

1. **Rewrite 2000 in terms of powers of 10**:
\[ 2000 = 2 \times 10^3 \]
Therefore,
\[ (2000)^{10} = (2 \times 10^3)^{10} \]

2. **Apply the power of a product rule**:
\[ (2000)^{10} = 2^{10} \times (10^3)^{10} \]
This simplifies to:
\[ 2^{10} \times 10^{30} \]

3. **Calculate \( 2^{10} \)**:
\[ 2^{10} = 1024 \]
So we can rewrite the equation as:
\[ 1024 \times 10^{30} = 1.024 \times 10^k \]

4. **Express 1024 in terms of 1.024**:
We can express \( 1024 \) as:
\[ 1024 = 1.024 \times 10^3 \]
Thus, we can rewrite the equation:
\[ 1.024 \times 10^3 \times 10^{30} = 1.024 \times 10^k \]

5. **Combine the powers of 10**:
\[ 1.024 \times 10^{33} = 1.024 \times 10^k \]

6. **Since the bases are equal, equate the exponents**:
\[ 33 = k \]

Thus, the value of \( k \) is \( 33 \).

### Final Answer:
\[ k = 33 \]