Simplify the surd below. Write your answer with a fractional exponent. 135√3

To simplify the surd $135\sqrt{3}$ and express it with a fractional exponent, follow these steps:

1. Break Down the Problem

   • Identify the components of the expression: $135$ and $\sqrt{3}$.
   • Recognize that $\sqrt{3}$ can be expressed as $3^{1/2}$.

2. Relevant Concepts

   • The expression $\sqrt{3}$ is equivalent to $3^{1/2}$.
   • The expression $135$ can be broken down into its prime factors.

3. Analysis and Detail

   • Factorize $135$ into its prime factors: $135 = 3^3 \times 5$.
   • Rewrite the expression using these factors: $135\sqrt{3} = 3^3 \times 5 \times 3^{1/2}$.
   • Combine the powers of 3: $3^3 \times 3^{1/2} = 3^{3 + 1/2} = 3^{7/2}$.
   • Therefore, the expression becomes $3^{7/2} \times 5$.

4. Verify and Summarize

   • Verify the calculations: $3^3 = 27$ and $\sqrt{3} = 3^{1/2}$.
   • Combine these to ensure the expression is simplified correctly.

Final Answer

The simplified form of the surd $135\sqrt{3}$ with a fractional exponent is $5 \times 3^{7/2}$.