https://file.aiquickdraw.com/mathbot/file/1727352084512oxuvp2gb.jpg

The problem in the image is asking for the value of \( 7^a \) in terms of \( b \), given the equation:

\[
2^a = 14^{a + b - 1}
\]

Let's start by breaking it down and solving step by step.

### Step 1: Express \( 14 \) as a product of primes
Since \( 14 = 2 \times 7 \), we can rewrite the equation \( 14^{a + b - 1} \) as:

\[
2^a = (2 \times 7)^{a + b - 1}
\]

This simplifies to:

\[
2^a = 2^{a + b - 1} \times 7^{a + b - 1}
\]

### Step 2: Compare the powers of 2 on both sides
On the left-hand side, we have \( 2^a \), and on the right-hand side, the factor involving \( 2 \) is \( 2^{a + b - 1} \). To satisfy this equation, the powers of 2 must be equal. Therefore, we equate the powers of 2:

\[
a = a + b - 1
\]

Simplifying this equation:

\[
0 = b - 1 \quad \Rightarrow \quad b = 1
\]

### Step 3: Find \( 7^a \)
We are asked to find \( 7^a \) in terms of \( b \). Since \( b = 1 \), we substitute this back into the original equation:

\[
2^a = 14^{a + 1 - 1} = 14^a = (2 \times 7)^a = 2^a \times 7^a
\]

Dividing both sides by \( 2^a \) gives:

\[
1 = 7^a
\]

Thus, we find:

\[
7^a = 1
\]

### Final Answer:
The value of \( 7^a \) is \( 1 \) when \( b = 1 \).

Would you like further clarification or additional steps explained?

Here are five questions to expand on this concept:
1. How do you solve exponential equations when variables are in the exponent?
2. What are some common techniques for dealing with exponential equations involving different bases?
3. Can you generalize this problem for any constant besides 14?
4. How does logarithmic manipulation help in solving exponential equations?
5. How can we apply this concept to real-world growth or decay problems?

**Tip:** When solving exponential equations, always look for ways to express numbers as powers of the same base to simplify comparisons.

For the real numbers a and b, given the equation 2^a = 14^(a+b-1), find 7^a in terms of b.

In this problem, we solve the equation 2^a = 14^(a+b-1) and find the value of 7^a in terms of b. This example covers exponential equations, prime factorization, and exponent comparison techniques, ideal for students in Grades 10-12.

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