 solve each exponential equation 7^2x+3= 49^8-x

To solve the exponential equation:

\[
7^{2x+3} = 49^{8 - x}
\]

### Step 1: Rewrite the Equation with a Common Base
Notice that \(49\) can be written as \(7^2\). Therefore, rewrite \(49^{8 - x}\) as \((7^2)^{8 - x}\).

This gives:
\[
7^{2x + 3} = (7^2)^{8 - x}
\]

### Step 2: Apply the Power Rule
Using the rule \((a^m)^n = a^{m \cdot n}\), we can rewrite the equation as:

\[
7^{2x + 3} = 7^{2(8 - x)}
\]

### Step 3: Set Exponents Equal to Each Other
Since the bases are the same, we can set the exponents equal to each other:

\[
2x + 3 = 2(8 - x)
\]

### Step 4: Simplify the Equation
Distribute on the right side:

\[
2x + 3 = 16 - 2x
\]

Now, add \(2x\) to both sides to combine like terms:

\[
4x + 3 = 16
\]

Subtract \(3\) from both sides:

\[
4x = 13
\]

Finally, divide by \(4\):

\[
x = \frac{13}{4} = 3.25
\]

### Solution
The solution is:
\[
x = \frac{13}{4} \quad \text{or} \quad x = 3.25
\]

Would you like further clarification on any step, or do you have additional questions?

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Here are 5 related questions to deepen your understanding:

1. How would you solve the equation if it was \(7^{2x+3} = 49^{x-8}\)?
2. What if the base on the right side was different, such as \(36^{8 - x}\)?
3. How does the solution process change if the exponents involve subtraction or division?
4. Can you apply a similar approach to solve \(3^{4x+1} = 81^{2 - x}\)?
5. How would you check your answer to verify that it's correct?

**Tip:** When solving exponential equations, try expressing all terms with the same base—it often simplifies the problem significantly.

solve each exponential equation 7^2x+3= 49^8-x

This guide explains how to solve the exponential equation 7^(2x+3) = 49^(8-x) by rewriting the equation with a common base and setting exponents equal. Learn to simplify and solve using key exponential concepts and properties, suitable for students in Grades 10-12.

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