The expression provided in the image is:

$4^{\frac{\sqrt{3}}{2}} \cdot \left( \frac{1}{8} \right)^{\sqrt{27}} = \left(16^{\sqrt[5]{3}}\right)^{-2}$

Let's solve this step-by-step.

Step 1: Simplify the Left-Hand Side

The left-hand side of the equation is:

$4^{\frac{\sqrt{3}}{2}} \cdot \left( \frac{1}{8} \right)^{\sqrt{27}}$

We can rewrite each term using exponents:

1. $4^{\frac{\sqrt{3}}{2}} = (2^2)^{\frac{\sqrt{3}}{2}} = 2^{2 \cdot \frac{\sqrt{3}}{2}} = 2^{\sqrt{3}}$

2. $\left(\frac{1}{8}\right)^{\sqrt{27}} = \left(8^{-1}\right)^{\sqrt{27}} = 8^{-\sqrt{27}}$

Since $8 = 2^3$, we can rewrite it as:

$8^{-\sqrt{27}} = (2^3)^{-\sqrt{27}} = 2^{-3\sqrt{27}}$

Therefore, the left-hand side becomes:

$2^{\sqrt{3}} \cdot 2^{-3\sqrt{27}} = 2^{\sqrt{3} - 3\sqrt{27}}$

Step 2: Simplify the Right-Hand Side

The right-hand side of the equation is:

$\left(16^{\sqrt[5]{3}}\right)^{-2}$

We know $16 = 2^4$, so:

[ 16^{\sqrt[5]{3}} = (2^4)^{\sqrt[5]{3}} = 2^{4\sqrt[5]{3}} ]

Raising it to the power of $-2$:

[ \left(2^{4\sqrt[5]{3}}\right)^{-2} = 2^{-8\sqrt[5]{3}} ]

Step 3: Equating the Exponents

Now, we equate the exponents from both sides:

$\sqrt{3} - 3\sqrt{27} = -8\sqrt[5]{3}$

Step 4: Solving for the Expression

To solve this equation, we need to simplify further:

• Recall that $\sqrt{27} = \sqrt{3^3} = 3\sqrt{3}$.
• Also, $\sqrt[5]{3}$ remains as is.

So, substituting these into the equation:

$\sqrt{3} - 3(3\sqrt{3}) = -8\sqrt[5]{3}$

This simplifies to:

$\sqrt{3} - 9\sqrt{3} = -8\sqrt[5]{3}$

$-8\sqrt{3} = -8\sqrt[5]{3}$

Divide both sides by $-8$:

$\sqrt{3} = \sqrt[5]{3}$

Conclusion

The equation holds true when $\sqrt{3} = \sqrt[5]{3}$. This is a unique scenario where both sides are equal.

If you need further details or have any questions, feel free to ask!

Related Questions:

1. How do you simplify expressions with fractional exponents?
2. What is the process for converting radical expressions to exponential form?
3. How can you solve equations with different roots, like square roots and fifth roots?
4. What are some strategies for solving equations with nested exponents?
5. How can properties of exponents help in solving complex equations?

Tip:

When dealing with radicals and exponents, always try to rewrite them in their exponential form to make simplification easier.