(2^((logn)^(1/2)))(n) covert the expression to the power of n

We are given the expression:

\[
2^{(\log n)^{\frac{1}{2}}} \cdot n
\]

To express this in terms of a single power of \( n \), let's follow these steps:

### Step 1: Write \( n \) as a power of 2

Since we know that \( n = 2^{\log_2 n} \), we can rewrite the \( n \) term as:

\[
n = 2^{\log_2 n}
\]

Now, our expression becomes:

\[
2^{(\log n)^{\frac{1}{2}}} \cdot 2^{\log_2 n}
\]

### Step 2: Combine the exponents

When multiplying terms with the same base (in this case, base 2), we add their exponents:

\[
2^{(\log n)^{\frac{1}{2}} + \log_2 n}
\]

### Final Expression:

Thus, the expression is now:

\[
2^{(\log n)^{\frac{1}{2}} + \log_2 n}
\]

This is the simplified expression in terms of the power of \( n \).

Would you like more details or have any questions?

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Here are 5 related questions to explore:
1. How do logarithms work when combined with exponents?
2. What is the change of base formula for logarithms?
3. How can we simplify more complex expressions involving powers of \( n \)?
4. How does \( n = 2^{\log_2 n} \) hold mathematically?
5. How do logarithmic and exponential functions relate in calculus?

**Tip:** When multiplying terms with the same base, always add their exponents to simplify the expression.

2^((logn)^(1/2))(n) convert the expression to the power of n

Explore how to simplify the expression 2^((logn)^(1/2))n and express it as a single power of n. This solution covers logarithmic identities, exponentiation rules, and step-by-step algebraic manipulation. Suitable for students in Grades 10-12.

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