https://file.aiquickdraw.com/mathbot/file/1726297177409hyfvcu96.jpg

The expression in the image can be interpreted as:

\[
\left( \frac{2\sqrt{27} \times 8\sqrt{75}}{4\sqrt{48}} \right)^{\frac{\sqrt{3}}{5}}
\]

Let's break it down step by step:

### Step 1: Simplifying the square roots
- \( \sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3} \)
- \( \sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3} \)
- \( \sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3} \)

### Step 2: Substituting these simplified terms back into the expression:
\[
\left( \frac{2(3\sqrt{3}) \times 8(5\sqrt{3})}{4(4\sqrt{3})} \right)^{\frac{\sqrt{3}}{5}}
\]

### Step 3: Simplifying the products:
- Numerator: \( 2 \times 3\sqrt{3} \times 8 \times 5\sqrt{3} = 240(\sqrt{3})^2 = 240 \times 3 = 720 \)
- Denominator: \( 4 \times 4\sqrt{3} = 16\sqrt{3} \)

So, the expression becomes:
\[
\left( \frac{720}{16\sqrt{3}} \right)^{\frac{\sqrt{3}}{5}} = \left( 45\frac{1}{\sqrt{3}} \right)^{\frac{\sqrt{3}}{5}} = \left( \frac{45}{\sqrt{3}} \right)^{\frac{\sqrt{3}}{5}}
\]

### Step 4: Rationalizing the denominator:
Multiply the numerator and denominator by \( \sqrt{3} \) to rationalize the denominator:
\[
\frac{45\sqrt{3}}{3} = 15\sqrt{3}
\]

So the expression becomes:
\[
\left( 15\sqrt{3} \right)^{\frac{\sqrt{3}}{5}}
\]

This is the simplified expression raised to the power of \( \frac{\sqrt{3}}{5} \).

Let me know if you'd like further details or questions on how this was handled!

### Relative Questions:
1. How do you simplify radicals like \( \sqrt{48} \)?
2. What is the process of rationalizing the denominator?
3. How do exponents work when they are in fractional form like \( \frac{\sqrt{3}}{5} \)?
4. What is the meaning of square root exponents in terms of logarithms?
5. How does one handle radicals in the numerator and denominator of an expression?

### Tip:
Always remember to simplify square roots by factoring them into their prime components to make calculations easier.

Simplify the mathematical expression provided in the image.

Learn to simplify a complex expression involving radicals, exponents, and rationalization. This problem focuses on square root simplification, multiplication of radicals, exponent rules, and rationalizing the denominator. Suitable for Grades 9-11.

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