https://file.aiquickdraw.com/mathbot/file/1726527777638x699gzml.jpg

The image shows three exponential equations. Let me solve them one by one.

### 1. \( 7^{x+4} = 49^{2x-1} \)

We can rewrite \(49\) as \(7^2\):
\[
49^{2x-1} = (7^2)^{2x-1} = 7^{2(2x-1)} = 7^{4x - 2}
\]

Now the equation becomes:
\[
7^{x+4} = 7^{4x-2}
\]

Since the bases are the same, we equate the exponents:
\[
x + 4 = 4x - 2
\]

Solving for \(x\):
\[
4 + 2 = 4x - x \implies 6 = 3x \implies x = 2
\]

### 2. \( \left( \frac{1}{2} \right)^{2x} = 2^{3-x} \)

We rewrite \( \left( \frac{1}{2} \right)^{2x} \) as \( 2^{-2x} \):
\[
2^{-2x} = 2^{3 - x}
\]

Since the bases are the same, we equate the exponents:
\[
-2x = 3 - x
\]

Solving for \(x\):
\[
-2x + x = 3 \implies -x = 3 \implies x = -3
\]

### 3. \( 4^{x+2} = 8^{2x} \)

We rewrite \(4\) and \(8\) with base 2:
\[
4 = 2^2 \quad \text{and} \quad 8 = 2^3
\]

Thus, the equation becomes:
\[
(2^2)^{x+2} = (2^3)^{2x}
\]

Simplifying both sides:
\[
2^{2(x+2)} = 2^{6x}
\]

Equate the exponents:
\[
2(x + 2) = 6x
\]

Simplifying:
\[
2x + 4 = 6x \implies 4 = 4x \implies x = 1
\]

### Final Answers:
1. \( x = 2 \)
2. \( x = -3 \)
3. \( x = 1 \)

Let me know if you want any further explanations or details.

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Here are 5 related questions for practice:
1. Solve \( 5^{x+1} = 25^{2x-3} \).
2. Solve \( 3^{2x} = 9^{x+1} \).
3. Solve \( 2^{x+5} = 16^{x-2} \).
4. Solve \( (1/3)^{2x} = 3^{x-1} \).
5. Solve \( 6^{x+2} = 36^{x-3} \).

**Tip**: Always try to express the numbers in exponential equations with the same base for easier simplification.

Answer the following exponential equations: 1. 7^(x+4) = 49^(2x-1), 2. (1/2)^(2x) = 23^(3-x), 3. 4^(x+2) = 8^(2x)

Learn how to solve three exponential equations step by step: 7^(x+4) = 49^(2x-1), (1/2)^(2x) = 23^(3-x), and 4^(x+2) = 8^(2x). Explore key algebraic concepts such as exponent rules and converting terms to the same base. Suitable for high school students in Grades 10-12.

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