Two finite sets have `m` and `3` elements respectively. The total number of subsets of first set is `56` more than the total number of subsets of the second set. The value of `m` is ___________

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the problem
We have two finite sets, one with `m` elements and the other with `3` elements. We know that the number of subsets of the first set is `56` more than the number of subsets of the second set.

### Step 2: Write the formula for the number of subsets
The number of subsets of a set with `n` elements is given by the formula \(2^n\).

### Step 3: Calculate the number of subsets for both sets
- For the first set (with `m` elements), the number of subsets is \(2^m\).
- For the second set (with `3` elements), the number of subsets is \(2^3 = 8\).

### Step 4: Set up the equation based on the problem statement
According to the problem, the number of subsets of the first set is `56` more than the number of subsets of the second set. This can be expressed as:
\[ 2^m = 8 + 56 \]

### Step 5: Simplify the equation
Now simplify the right side of the equation:
\[ 2^m = 64 \]

### Step 6: Solve for `m`
We know that \(64\) can be expressed as a power of \(2\):
\[ 64 = 2^6 \]
Thus, we can set the exponents equal to each other:
\[ m = 6 \]

### Final Answer
The value of \(m\) is **6**.

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