Simplify the surd below. Write your answer with a fractional exponent. √45

To simplify the surd $\sqrt{45}$ and express it with a fractional exponent, follow these steps:

1. Break Down the Problem

   • Identify the prime factors of 45.
   • Express the square root in terms of these factors.

2. Relevant Concepts

   • Use the property of square roots: $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$.
   • Recognize that $\sqrt{a^2} = a$.

3. Analysis and Detail

   • Factor 45 into prime factors: $45 = 3^2 \times 5$.
   • Apply the square root property: $\sqrt{45} = \sqrt{3^2 \times 5} = \sqrt{3^2} \times \sqrt{5}$.
   • Simplify: $\sqrt{3^2} = 3$, so $\sqrt{45} = 3 \times \sqrt{5}$.

4. Verify and Summarize

   • Verify the simplification: $3 \times \sqrt{5}$ is the simplest form.
   • Express $\sqrt{5}$ with a fractional exponent: $\sqrt{5} = 5^{1/2}$.

Final Answer

The simplified form of $\sqrt{45}$ with a fractional exponent is $3 \times 5^{1/2}$.