https://file.aiquickdraw.com/mathbot/file/1727363194430euzn6qnw.jpg

The equation shown in the image is:

\[
3^{3x} \times \left(\frac{4}{9}\right)^{4 - 3x} = 3
\]

### Let's solve this step by step.

1. **Rewrite \( \left(\frac{4}{9}\right)^{4 - 3x} \) using exponent laws:**
   \[
   \left(\frac{4}{9}\right)^{4 - 3x} = \frac{4^{4 - 3x}}{9^{4 - 3x}} = \frac{4^{4 - 3x}}{3^{2(4 - 3x)}}
   \]
   So the equation becomes:
   \[
   3^{3x} \times \frac{4^{4 - 3x}}{3^{2(4 - 3x)}} = 3
   \]

2. **Combine the terms with base 3:**
   The expression simplifies by combining the powers of 3:
   \[
   \frac{3^{3x}}{3^{8 - 6x}} \times 4^{4 - 3x} = 3
   \]
   Simplify the powers of 3:
   \[
   3^{3x - (8 - 6x)} \times 4^{4 - 3x} = 3
   \]
   Simplifying the exponent:
   \[
   3^{9x - 8} \times 4^{4 - 3x} = 3^1
   \]

3. **Equating powers of 3:**
   Since the equation involves powers of 3, we can compare the exponents on both sides:
   \[
   9x - 8 = 1
   \]
   Solve for \( x \):
   \[
   9x = 9 \quad \Rightarrow \quad x = 1
   \]

Thus, the solution to the equation is:

\[
x = 1
\]

Would you like any more details or clarification on the steps? Here are five related questions to expand the concept:

1. How do you simplify fractional powers with different bases like \( \left(\frac{4}{9}\right)^n \)?
2. Why are powers with the same base compared directly in exponential equations?
3. What happens if the equation involves different bases, like \( 2^x \times 3^{2x} = 6 \)?
4. How do logarithmic methods apply to solving exponential equations?
5. Can you solve \( 2^{x + 1} \times 4^{2x - 3} = 32 \)?

**Tip:** When solving exponential equations, always try to express terms in the equation using the same base to simplify the comparison.

Solve the equation 3^(3x) * (4/9)^(4 - 3x) = 3.

Step-by-step solution of the exponential equation 3^(3x) * (4/9)^(4 - 3x) = 3. Learn how to apply exponent rules to solve for x. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation