Given `log_(10 ) x = 2a and log_(10) y = (b)/(2)`.
Write ` 10 ^(a) ` in terms of x.

Answer

To solve the problem, we need to express \( 10^a \) in terms of \( x \) given that \( \log_{10} x = 2a \).

### Step-by-step Solution:

1. **Start with the given equation:**
\[ \log_{10} x = 2a \]

2. **Rearrange the equation to express \( a \):**
To isolate \( a \), we can divide both sides by 2:
\[ \frac{1}{2} \log_{10} x = a \]

3. **Use the properties of logarithms:**
According to the properties of logarithms, we can express \( a \) as:
\[ a = \log_{10} (x^{1/2}) \]
This follows from the property \( k \cdot \log_b a = \log_b (a^k) \).

4. **Express \( 10^a \):**
Now, we can express \( 10^a \) using the definition of logarithms:
\[ 10^a = 10^{\log_{10} (x^{1/2})} \]

5. **Apply the inverse property of logarithms:**
By the property of logarithms, \( 10^{\log_{10} b} = b \):
\[ 10^a = x^{1/2} \]

6. **Final expression:**
Therefore, we can write:
\[ 10^a = \sqrt{x} \]

### Final Answer:
\[ 10^a = \sqrt{x} \]