https://file.aiquickdraw.com/mathbot/file/173047795854286t5bxa0.jpg

Let's solve this step by step. The problem is:

\[
\frac{9^{\frac{2}{3}} \cdot 27^{\frac{3}{2}}}{81^{\frac{1}{2}}} = 3^x
\]

### Step 1: Rewrite Each Base in Terms of 3

To simplify, let’s express each base (9, 27, and 81) as powers of 3.

- \( 9 = 3^2 \)
- \( 27 = 3^3 \)
- \( 81 = 3^4 \)

### Step 2: Substitute and Simplify Each Term

Now, substitute these values:

1. \( 9^{\frac{2}{3}} = (3^2)^{\frac{2}{3}} = 3^{2 \cdot \frac{2}{3}} = 3^{\frac{4}{3}} \)
2. \( 27^{\frac{3}{2}} = (3^3)^{\frac{3}{2}} = 3^{3 \cdot \frac{3}{2}} = 3^{\frac{9}{2}} \)
3. \( 81^{\frac{1}{2}} = (3^4)^{\frac{1}{2}} = 3^{4 \cdot \frac{1}{2}} = 3^2 \)

Now the equation becomes:

\[
\frac{3^{\frac{4}{3}} \cdot 3^{\frac{9}{2}}}{3^2} = 3^x
\]

### Step 3: Combine the Exponents on the Left Side

Using the property \( a^m \cdot a^n = a^{m+n} \):

\[
3^{\frac{4}{3} + \frac{9}{2} - 2} = 3^x
\]

Now, we need to find a common denominator to simplify the exponents. The least common denominator of 3, 2, and 1 is 6.

Rewrite each exponent:

- \( \frac{4}{3} = \frac{8}{6} \)
- \( \frac{9}{2} = \frac{27}{6} \)
- \( 2 = \frac{12}{6} \)

So:

\[
3^{\frac{8}{6} + \frac{27}{6} - \frac{12}{6}} = 3^x
\]

Combine the terms in the exponent:

\[
3^{\frac{8 + 27 - 12}{6}} = 3^x
\]

\[
3^{\frac{23}{6}} = 3^x
\]

### Step 4: Set the Exponents Equal to Each Other

Since the bases are the same, we can set the exponents equal to each other:

\[
x = \frac{23}{6}
\]

### Final Answer

\[
x = \frac{23}{6}
\]

Would you like more details on any of these steps? 

---

Here are some related questions to deepen your understanding:

1. How would you approach a similar problem if the bases were different?
2. What are other ways to simplify fractional exponents in complex expressions?
3. How do the rules of exponents apply when dividing terms with fractional exponents?
4. Why is it helpful to express each term with the same base in exponential equations?
5. What would happen if a different method were used to solve this equation?

**Tip:** When working with exponents, always look for ways to rewrite numbers using the same base. This often simplifies calculations and comparisons.

Solve for x in the equation: (9^(2/3) * 27^(3/2)) / (81^(1/2)) = 3^x

Learn how to solve for x in the exponential equation (9^(2/3) * 27^(3/2)) / (81^(1/2)) = 3^x by rewriting terms with the same base and simplifying fractional exponents. Ideal for high school students studying advanced algebra and exponent rules.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation