Square Root Of Numbers With Decimals

To extract the square root of a number containing whole numbers and decimal numbers, we use the same procedure as to extract the square root of the whole numbers. However, the difference is in the way in which the periods will be divided, since in the case of numbers with integers and decimals, the decimal point will be our reference point. Thus, we will form the periods of the integers by separating two by two, from the decimal point to the left. In decimals, we will form the periods by forming the pairs of the decimal point to the right.

Steps to take the square root of numbers with decimals:

So, for example, if we have the following number:

25473.117609

STEP 1. We start by dividing the periods from the decimal point, the integers to the left, the decimals to the right:

STEP 2: We calculate the root closest to the first period. In this case the first root is 1. We subtract it and we will have 1 as remainder.

STEP 3: we lower the next period, 54 and separate the last figure. On the result side, we double the root, which will give us 2. We divide 15 by 2, which gives us 7, however, it is a very high number, so we will take 5 as the value. We add the 5 in the root; We also add it to the double of the root, and we multiply this number by 5, which gives us 125. We subtract 125 from 154, which gives us a remainder of 29.

STEP 4: Now we lower the next period (73) and separate the last figure, with which we will have 297'3. On the right side we double the root 15, which gives us 30 as a result. We divide 297 by 30, and it gives us 9. We add the 9 in the result of the root, to the double of the root, and this last result we multiply by 9. 309 times 9 gives us 2781. We subtract it from 2973 and it gives us 192 as a remainder.

STEP 5: We lower the next period, which is 11. Note that this period is the first after the decimal point, so in the result of the root, we will put the decimal point from this result. We separate the last figure, with what we will have: 1921'1. On the right side we double the root: 159, which gives us 318. We divide 1921 by 318, which gives us 6. We add the decimal point and the number 6 in the root; We also add it to the double of the root, and multiply by 6, which gives us 19116, which we will subtract from 19211, leaving us as a remainder of 95.

STEP 6: We lowered the following period: 57. We separate the last number and we will have 957'6. On the right side we double the root 1596 which gives us 3192. If we try to divide 957 by 3192 the result is less than 1, so in this case, the next number in the root will be 0.

STEP 7: We lower the following period: 09. Now, separating the last number, we will have 95,760'9. To our right, we double the root 15960, which gives us 31920. We divide 95,760 by 31,920, which gives us 3. We add 3 to our root, also twice the root, and multiply it by 3. The result is 957609, so when you perform the subtraction, the result is exact.