To solve the equation:

$\left( \sqrt[3]{6} \right)^{2x} = \left( \sqrt{6} \right)^{x + 6}$

First, let's express the radicals as exponents:

• $\sqrt[3]{6} = 6^{1/3}$
• $\sqrt{6} = 6^{1/2}$

The equation then becomes: $\left( 6^{1/3} \right)^{2x} = \left( 6^{1/2} \right)^{x+6}$

Using the power rule $(a^m)^n = a^{m \cdot n}$, we can simplify both sides:

$6^{\frac{2x}{3}} = 6^{\frac{x + 6}{2}}$

Since the bases are the same (both are powers of 6), we can set the exponents equal to each other:

$\frac{2x}{3} = \frac{x + 6}{2}$

Now, solve for $x$ by cross-multiplying:

$2 \times 2x = 3 \times (x + 6)$ $4x = 3x + 18$ $4x - 3x = 18$ $x = 18$

Thus, $x = 18$.

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

Related Questions:

1. How do you simplify powers with fractional exponents?
2. What are the steps for solving equations with the same base?
3. How do you use logarithms to solve exponential equations?
4. Can you solve exponential equations without common bases? How?
5. How does the power rule of exponents work in equations?

Tip: Always check if you can simplify the equation by expressing radicals as exponents to make solving easier!