The given equation is:

$8 = \left( \sqrt{2} \right)^x$

To solve for $x$, follow these steps:

Step 1: Rewrite 8 in terms of powers of 2.

$8 = 2^3$

Step 2: Express $\sqrt{2}$ as $2^{1/2}$.

$\left( \sqrt{2} \right)^x = \left( 2^{1/2} \right)^x = 2^{x/2}$

Now, the equation becomes: $2^3 = 2^{x/2}$

Step 3: Since the bases are the same (base 2), equate the exponents:

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

Step 4: Solve for $x$.

Multiply both sides of the equation by 2: $x = 6$

Final Answer:

$x = 6$

Would you like more details on this solution or any related concepts?

Here are 5 related questions you could explore:

1. How do you solve logarithmic equations involving square roots?
2. What happens when the base of both sides is different?
3. How can we graph exponential and logarithmic functions?
4. How do you solve $a^x = b$ when $a$ and $b$ are different powers of 2?
5. What are some real-world applications of exponential equations?

Tip: When solving equations with exponents, try to express both sides with the same base whenever possible—it simplifies the solving process significantly!