Find the value of p, if 25 power to(p – 1) + 100 = 5 power to(2p – 1).

We are given the equation:

\[
25^{(p-1)} + 100 = 5^{(2p-1)}
\]

### Step 1: Express powers of 25 and 100 in terms of base 5.
We know that:
\[
25 = 5^2 \quad \text{and} \quad 100 = 5^2 \times 4
\]
Using this, we rewrite the equation as:

\[
(5^2)^{(p-1)} + 100 = 5^{(2p-1)}
\]

This simplifies to:

\[
5^{2(p-1)} + 100 = 5^{(2p-1)}
\]

\[
5^{2p-2} + 100 = 5^{2p-1}
\]

### Step 2: Isolate the terms involving powers of 5.
Now, subtract \( 5^{2p-2} \) from both sides:

\[
5^{2p-2}(1 + 100 \cdot 5^{-(2p-2)})
\]

###

Find the value of p, if 25 power to (p – 1) + 100 = 5 power to (2p – 1).

Learn how to solve the exponential equation 25^(p-1) + 100 = 5^(2p-1) by expressing powers of 25 and 100 in terms of base 5. This step-by-step solution demonstrates key algebraic concepts, including exponent rules and algebraic manipulation. Suitable for students in Grades 9-11.

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