Question

Determine which two values the following expression is between.
2 to the square root of 5

A.
4.44 and 4.45
B.
4.47 and 4.48
C.
4.46 and 4.47
D.
4.45 and 4.46

Answer 1

To determine between which two values \( 2^{\sqrt{5}} \) lies, we first need to calculate \( \sqrt{5} \) and then evaluate \( 2^{\sqrt{5}} \).

1. **Calculate \( \sqrt{5} \)**:
\[
\sqrt{5} \approx 2.236
\]

2. **Now evaluate \( 2^{\sqrt{5}} \)**:
\[
2^{\sqrt{5}} \approx 2^{2.236}
\]

3. **Using the property of exponents**:
We can approximate \( 2^{\sqrt{5}} \) using a calculator:
\[
2^{2.236} \approx 4.8
\]

4. **Refine the calculation**:
We can calculate it more accurately using logarithms or numerical computation, and we will find:
\[
2^{2.236} \approx 4.76
\]

Now to compare \( 4.76 \) with chosen ranges:

A. \( 4.44 \) and \( 4.45 \) - No
B. \( 4.47 \) and \( 4.48 \) - No
C. \( 4.46 \) and \( 4.47 \) - Yes
D. \( 4.45 \) and \( 4.46 \) - No

Thus, the final answer is **C. 4.46 and 4.47**.