Square root of a square number

Square root of a square number

The square root of a square number is the opposite of the second power of the number. Finding the square root of a square number is the same as finding the base of the second power number. The square root of a square number is represented by the symbol √. The symbol √ means that a number is raised to the half power, that is $√a = a^{1/2}$.

Finding the square root of a square number with not more than three digits

The computation of a square root of a square number is done by using different methods including tree diagrams, factorization and grouping of digits.

Example 1

Find the square root of 256 by using a tree diagram.

Solution

Use the following steps:

1. Draw a tree diagram and obtain all prime factors

2. Write the square root of 256 as a product of prime factors:

$√256 = √(2 × 2 × 2 × 2 × 2 × 2 × 2 × 2)$

3. Arrange in pairs the product of the same prime factors:

$√256 = √2 × 2 × √2 × 2 × √2 × 2 × √2 × 2$

4. Take one factor from each pair of prime factors, and then multiply. The product of these factors will be the square root of 256. That is, $√256 = 2 × 2 × 2 × 2 = 16$.

Therefore, the square root of 256 is 16.

Example 2

Find the square root of 196 using the method of prime factors.

Solution

Use the following steps:

1. Divide 196 as shown below in order to get all its prime factors:

| 2 | 196 | | 2 | 98 | | 7 | 49 | | 7 | 7 | | | 1 |

2. Write the square root of 196 as a product of its prime factors:

$√196 = √(2 × 2 × 7 × 7)$

3. Arrange in pairs the product of the same prime factors:

$√196 = √2 × 2 × √7 × 7$

4. Take one factor from each pair of prime factors, and then multiply them. The product of these factors is the square root of 196.

Thus, $√196 = 2 × 7 = 14$.

Therefore, the square root of 196 is 14.

Example 3

Find the square root of 625 by grouping digits.

Solution

Use the following steps:

1. Group the digits into pairs from right.	√ 6 25
2. Find a number which when multiplied by itself the product is 6 or it approaches 6 and does not exceed 6. The required number is 2 since 2 × 2 = 4.
3. Write 2 on top of 6 in the answer's position. Also, write 2 to the left as a divisor.	2 √ 6 25
4. Multiply 2 of the part of the answer by 2 of the part of the divisor, 2 × 2 = 4, and then subtract their product from 6.	2 2√ 6 25 − 4 2
5. Bring down the next two digits grouped from right and write them in front of 2 to get 225. Also, add 2 of the divisor and 2 of part of the answer to get a new divisor 2 + 2 = 4. Write 4 on the left of 225.	2 2√ 6 25 − 4 4 2 25
6. Find a number to be written to the right of 2 in the answer position. The same number should be written in front of a new divisor. This number is multiplied by a new divisor to get 225 or a product close to 225. The required number is 5. The new divisor becomes 45 and the value in the answer will be 25. Multiply 45 by 5 of part of the answer, 5 × 45 = 225. Write 225 below 225, and then subtract to get 0.	2 5 2√ 6 25 − 4 45 2 25 − 2 25
Therefore, the square root of 625 is 25.