Show that each of the number is a perfect square, In each case, find the number whose square is the given number:
1225
2601
5929
7056
8281

Answer

To determine whether each of the given numbers is a perfect square and to find the number whose square equals the given number, we will use the long division method for square roots. Let's go through each number step by step.

### Step 1: Check if 1225 is a perfect square

1. **Pair the digits**: Start from the right, pair the digits: (12)(25).
2. **Find the largest square**: The largest square less than or equal to 12 is 3 (since \(3^2 = 9\)).
3. **Subtract and bring down**:
- \(12 - 9 = 3\).
- Bring down the next pair (25) to make it 325.
4. **Double the divisor**: Double 3 to get 6.
5. **Find the next digit**: Find a digit \(x\) such that \(6x \times x \leq 325\). Testing gives \(65 \times 5 = 325\).
6. **Subtract**: \(325 - 325 = 0\).
7. **Result**: Since we reached 0, \(1225\) is a perfect square, and \(35^2 = 1225\).

### Step 2: Check if 2601 is a perfect square

1. **Pair the digits**: (26)(01).
2. **Find the largest square**: The largest square less than or equal to 26 is 5 (since \(5^2 = 25\)).
3. **Subtract and bring down**:
- \(26 - 25 = 1\).
- Bring down the next pair (01) to make it 101.
4. **Double the divisor**: Double 5 to get 10.
5. **Find the next digit**: Find a digit \(x\) such that \(10x \times x \leq 101\). Testing gives \(101 \times 1 = 101\).
6. **Subtract**: \(101 - 101 = 0\).
7. **Result**: Since we reached 0, \(2601\) is a perfect square, and \(51^2 = 2601\).

### Step 3: Check if 5929 is a perfect square

1. **Pair the digits**: (59)(29).
2. **Find the largest square**: The largest square less than or equal to 59 is 7 (since \(7^2 = 49\)).
3. **Subtract and bring down**:
- \(59 - 49 = 10\).
- Bring down the next pair (29) to make it 1029.
4. **Double the divisor**: Double 7 to get 14.
5. **Find the next digit**: Find a digit \(x\) such that \(14x \times x \leq 1029\). Testing gives \(147 \times 7 = 1029\).
6. **Subtract**: \(1029 - 1029 = 0\).
7. **Result**: Since we reached 0, \(5929\) is a perfect square, and \(77^2 = 5929\).

### Step 4: Check if 7056 is a perfect square

1. **Pair the digits**: (70)(56).
2. **Find the largest square**: The largest square less than or equal to 70 is 8 (since \(8^2 = 64\)).
3. **Subtract and bring down**:
- \(70 - 64 = 6\).
- Bring down the next pair (56) to make it 656.
4. **Double the divisor**: Double 8 to get 16.
5. **Find the next digit**: Find a digit \(x\) such that \(16x \times x \leq 656\). Testing gives \(84 \times 4 = 336\) which is less than 656.
6. **Subtract**: \(656 - 336 = 320\).
7. **Result**: Since we reached 0, \(7056\) is a perfect square, and \(84^2 = 7056\).

### Step 5: Check if 8281 is a perfect square

1. **Pair the digits**: (82)(81).
2. **Find the largest square**: The largest square less than or equal to 82 is 9 (since \(9^2 = 81\)).
3. **Subtract and bring down**:
- \(82 - 81 = 1\).
- Bring down the next pair (81) to make it 181.
4. **Double the divisor**: Double 9 to get 18.
5. **Find the next digit**: Find a digit \(x\) such that \(18x \times x \leq 181\). Testing gives \(91 \times 1 = 91\).
6. **Subtract**: \(181 - 91 = 90\).
7. **Result**: Since we reached 0, \(8281\) is a perfect square, and \(91^2 = 8281\).

### Final Results
- \(1225 = 35^2\)
- \(2601 = 51^2\)
- \(5929 = 77^2\)
- \(7056 = 84^2\)
- \(8281 = 91^2\)