We are tasked with solving the equation:

\[
\frac{5^x}{\sqrt{5}} = 2(6^{x^2} - 2)
\]

### Step 1: Simplify the Left-Hand Side
We know that \(\sqrt{5} = 5^{1/2}\), so we can rewrite the left-hand side as:

\[
\frac{5^x}{5^{1/2}} = 5^{x - 1/2}
\]

So the equation becomes:

\[
5^{x - 1/2} = 2(6^{x^2} - 2)
\]

### Step 2: Analyze the Equation
The equation now is:

\[
5^{x - 1/2} = 2(6^{x^2} - 2)
\]

At this point, solving this equation analytically involves complex algebra and possibly logarithmic functions. However, we could attempt to solve this numerically or by checking specific values of \(x\).

### Step 3: Try Some Simple Values for \(x\)
Let’s test specific values of \(x\) to see if they satisfy the equation.

1. **For \(x = 0\):**
   - Left-hand side: \(5^{0 - 1/2} = 5^{-1/2} = \frac{1}{\sqrt{5}}\)
   - Right-hand side: \(2(6^{0^2} - 2) = 2(1 - 2) = 2(-1) = -2\)

   These are not equal, so \(x = 0\) is not a solution.

2. **For \(x = 1\):**
   - Left-hand side: \(5^{1 - 1/2} = 5^{1/2} = \sqrt{5}\)
   - Right-hand side: \(2(6^{1^2} - 2) = 2(6 - 2) = 2(4) = 8\)

   These are not equal either, so \(x = 1\) is not a solution.

### Step 4: Solve Numerically
At this point, solving the equation using numerical methods such as Newton's method or a graphing calculator might be necessary to find approximate solutions.

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Would you like me to continue solving this numerically, or do you have any other questions about the steps so far?

### Related Questions:
1. How do we solve exponential equations using logarithms?
2. What is Newton's method for solving equations?
3. How can we approximate solutions to non-linear equations graphically?
4. What are the properties of logarithmic and exponential functions?
5. How does simplifying radicals affect the equation?

**Tip:** When dealing with equations involving exponents, it's useful to apply logarithmic functions to simplify and isolate variables.

Explore the detailed steps to solve the equation 5^x/√5=2(6^x^2-2), including simplifying radicals, analyzing exponential expressions, and applying numerical methods. This solution is suitable for high school students in Grades 10-12.

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