To find the square root of 256 using the division method (also known as the long division method), follow these steps:

Step 1: Group the Digits

• Start by pairing the digits of 256 from right to left. In this case, since 256 has three digits, we group them as (2)(56).

Step 2: Find the Largest Number Whose Square is Less Than or Equal to the First Group

• Consider the first group, which is 2.
• The largest number whose square is less than or equal to 2 is 1 (since $1^2 = 1$).

Step 3: Subtract and Bring Down the Next Group

• Write 1 as the divisor and the quotient. Subtract $1^2$ from 2: $2 - 1 = 1$
• Now bring down the next group, which is 56, to get 156.

Step 4: Find a New Divisor

• Double the quotient (1) to get 2. Write 2 in front of a blank space (which will be filled with a digit from 0 to 9) to create the new divisor: 2_.
• Find the largest digit (X) such that when X is placed in the blank, 2X multiplied by X gives a product less than or equal to 156.
• The digit that works here is 6, since $26 \times 6 = 156$.

Step 5: Complete the Division

• Subtract $156$ from $156$ to get 0. The quotient is 16, which is the square root of 256.

Final Answer:

$\sqrt{256} = 16$

Would you like a more detailed explanation or have any other questions?

Here are some related questions:

1. How do you find square roots using the prime factorization method?
2. Can you explain the concept of perfect squares?
3. How does the long division method differ for non-perfect squares?
4. How do you approximate square roots using the division method?
5. What is the relationship between square roots and exponents?

Tip: When dealing with square roots of perfect squares, practicing the division method helps reinforce your understanding of basic arithmetic operations.