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

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

Example 1

Find the square root of 1,296 by using prime factors.

Solution

Use the following steps:

1. Divide in order to get the prime factors of 1,296.

   2 | 1296
   2 | 648
   2 | 324
   2 | 162
   3 | 81
   3 | 27
   3 | 9
   3 | 3
     | 1
   

2. Write the square root of 1,296 as a product of its prime factors.

   $\sqrt{1296} = \sqrt{2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3}$

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

   $\sqrt{1296} = \sqrt{2 \times 2} \times \sqrt{2 \times 2} \times \sqrt{3 \times 3} \times \sqrt{3 \times 3}$

4. Take one factor from each pair of prime factors, then multiply. The product of these factors is the square root of 1,296.

   $\sqrt{1296} = 2 \times 2 \times 3 \times 3 = 36$

   Therefore, the square root of 1,296 is 36.

Example 2

Find the square root of 21,904 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 21,904 as a product of its prime factors.

   $\sqrt{21904} = \sqrt{2 \times 2 \times 2 \times 2 \times 37 \times 37}$

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

   $\sqrt{21904} = \sqrt{2 \times 2} \times \sqrt{2 \times 2} \times \sqrt{37 \times 37}$

4. Take one factor from each pair of prime factors, then multiply. The product of these factors is the square root of 21,904.

   $\sqrt{21904} = 2 \times 2 \times 37 = 148$

   Therefore, the square root of 21,904 is 148.

Example 3

Find the square root of 948,676 by grouping the digits.

Solution

Use the following steps:

Step	Process
1. Group the digits into pairs from the right.	√94 86 76
2. Find a number which when multiplied by itself gives 94 or close to 94 and does not exceed 94. The required number is 9 since 9 × 9 = 81.
3. Write 9 on top of 94 in the answer's position. Also, write 9 to the left as a divisor.	9 9√94 86 76
4. Multiply 9 of part of the answer by 9 of part of a divisor: 9 × 9 = 81, then subtract their product from 94 to get 13.	9 9√94 86 76 −81 13
5. Bring down the next two digits and write them to the right of 13 to get 1,386. Add 9 of the divisor and 9 of the answer to get a new divisor, 9 + 9 = 18. Write 18 on the left of 1,386.	9 9√94 86 76 −81 13 86
6. Find a number to be written to the right of 9 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 1,386 or a product close to 1,386. The required number is 7. The new divisor becomes 187 and the value in the answer will be 97. Multiply 187 by 7 of part of the answer to get 1,309. Write 1,309 below 1,386 and subtract to get 77.	9 7 9√94 86 76 −81 13 86 7 7
7. Bring down the last two digits and write them in front of 77 to get 7,776. Add 187 of the divisor and 7 of part of the answer to get a new divisor, which is 194. Write 194 on the left of 7,776.	9 7 4 9 √94 86 76 −81 187 13 86 −13 09 194 77 76
8. Find a number to be written to the right of 97 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 7,776 or a product close to 7,776. The required number is 4. The new divisor becomes 1,944 and the value on the answer will be 974. Multiply 1,944 by 4 of part of the answer to get 7,776. Write 7,776 below 7,776 and subtract to get 0.	9 7 4 9 √94 86 76 −81 187 13 86 −13 09 1944 77 76 −77 76

Therefore, the square root of 948,676 is 974.