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.