https://file.aiquickdraw.com/mathbot/file/1725515027299dw112pcv.jpg

The equation in the image is:

\[
9^{\sqrt{x}} = \sqrt{27^x}
\]

Let's solve this step by step.

### Step 1: Rewrite the equation
First, simplify both sides of the equation using properties of exponents and radicals:

- The left side, \( 9^{\sqrt{x}} \), remains the same for now.
- The right side, \( \sqrt{27^x} \), can be rewritten as \( (27^x)^{1/2} \), or equivalently, \( 27^{x/2} \).

Thus, the equation becomes:

\[
9^{\sqrt{x}} = 27^{x/2}
\]

### Step 2: Express both sides with the same base
The base 9 and 27 can be rewritten as powers of 3:

- \( 9 = 3^2 \), so \( 9^{\sqrt{x}} = (3^2)^{\sqrt{x}} = 3^{2\sqrt{x}} \).
- \( 27 = 3^3 \), so \( 27^{x/2} = (3^3)^{x/2} = 3^{3x/2} \).

Now the equation becomes:

\[
3^{2\sqrt{x}} = 3^{3x/2}
\]

### Step 3: Set exponents equal
Since the bases are the same, we can set the exponents equal:

\[
2\sqrt{x} = \frac{3x}{2}
\]

### Step 4: Solve for \(x\)
Now, let's solve for \( x \).

1. Multiply both sides by 2 to eliminate the fraction:

\[
4\sqrt{x} = 3x
\]

2. Square both sides to remove the square root:

\[
(4\sqrt{x})^2 = (3x)^2
\]
\[
16x = 9x^2
\]

3. Rearrange the equation:

\[
9x^2 - 16x = 0
\]

4. Factor the quadratic equation:

\[
x(9x - 16) = 0
\]

So, the solutions are:

\[
x = 0 \quad \text{or} \quad x = \frac{16}{9}
\]

### Step 5: Verify the solutions
- For \( x = 0 \), substituting into the original equation gives \( 9^0 = \sqrt{27^0} \), which simplifies to \( 1 = 1 \), so \( x = 0 \) is a valid solution.
- For \( x = \frac{16}{9} \), substituting into the original equation confirms that both sides are equal, so \( x = \frac{16}{9} \) is also a valid solution.

### Final Answer:
The solutions are \( x = 0 \) and \( x = \frac{16}{9} \).

Do you want more details on any of the steps, or have any further questions?

Here are some related questions to consider:
1. How do you simplify radical expressions with exponents?
2. What are the common techniques for solving quadratic equations?
3. Can you explain how to convert bases in exponential equations?
4. What is the general approach for solving equations with radicals?
5. How do you factor quadratic expressions efficiently?

**Tip:** Always check if an equation can be simplified by expressing both sides with the same base. This is a common technique in solving exponential equations.

Learn how to solve the equation 9^√x = √27^x by simplifying radicals and exponents. This problem introduces exponentiation and quadratic techniques, suitable for Grades 9-11.

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