Let’s delve into the mathematical conundrum of the day. The problem at hand is expressed as follows: 3 to the power of M minus 2 to the power of M equals 65. Our mission is to determine the possible value of M.

Without further ado, let’s break down the solution step by step.

**Step 1: Simplifying the Equation**

We begin by writing the equation:

3^M – 2^M = 65

To simplify, we aim to introduce a common factor or variable. We decide to work with the exponents in this case. By squaring both sides of the equation, we can achieve this. This leads us to the following expression:

(3^M)^2 – (2^M)^2 = 65

**Step 2: Applying the Law of Indices**

Now, we apply the law of indices, which states that (a^m * b^n) equals a^(m * n). This allows us to rewrite our equation as:

(3^(M/2))^2 – (2^(M/2))^2 = 65

**Step 3: Introducing New Variables**

To make further progress, we introduce two new variables:

Let X = 3^(M/2) Let Y = 2^(M/2)

Our equation now becomes:

X^2 – Y^2 = 65

**Step 4: Using the Difference of Two Squares**

We recognize that X^2 – Y^2 is a difference of two squares, which can be factored as (a^2 – b^2) = (a + b)(a – b). Applying this rule, we rewrite the equation as:

(X + Y)(X – Y) = 65

**Step 5: Factoring 65**

Now, we look at the factors of 65, which are 5 and 13. We re-express 65 as 5 * 13 and substitute this into our equation:

(X + Y)(X – Y) = 5 * 13

**Step 6: Solving for X + Y and X – Y**

We have two equations now:

- X + Y = 13
- X – Y = 5

We can solve these two equations simultaneously. Adding equation 1 to equation 2, we get:

2X = 18

Dividing both sides by 2, we find:

X = 9

**Step 7: Finding Y**

We can substitute the value of X into either equation 1 or 2 to find Y. Let’s use equation 1:

9 + Y = 13

Subtracting 9 from both sides:

Y = 13 – 9 Y = 4

**Step 8: Finding the Value of M**

Recall our earlier substitutions: X = 3^(M/2) and Y = 2^(M/2).

Now, we have X = 9 and Y = 4. Let’s find the value of M:

For X: 3^(M/2) = 9

Express 9 as 3^2:

3^(M/2) = 3^2

Since the bases are the same, we can equate the exponents:

M/2 = 2

Now, solve for M:

M = 4

**Step 9: Verification**

To confirm that M = 4 satisfies our original equation, we substitute it back in:

3^4 – 2^4 = 65

Calculating:

81 – 16 = 65

Indeed, 65 equals 65, which verifies that M = 4 is the correct solution.

In conclusion, the value of M in the equation 3^M – 2^M = 65 is 4. I hope you found this solution insightful. If you learned something new from this method-only paid question, please share your thoughts in the comments below. Don’t hesitate to share this knowledge of mathematics with your family and friends.