Finding the square root of a large number or a decimal can feel intimidating, but the Long Division Method makes it systematic and manageable. Unlike standard division, this method focuses on “pairing” digits to find the root one step at a time.
Let’s look at two practical examples from our latest lesson.
In this example, we use the long division format to solve for a decimal number step-by-step.
Group the Digits: Starting from the decimal point, group the digits in pairs (bars) moving both left and right. Here, we get $\overline{1}\ \overline{10}.\overline{25}$.
First Digit: Find the largest square less than or equal to the first group (1). That is $1 \times 1 = 1$. Subtract and bring down the next pair (10).
Double the Quotient: Double your current answer (1) to get 2. We need a digit ‘$x$‘ such that $2x \times x$ is less than or equal to 10. Since $21 \times 1 = 21$ (too big), we use 0.
The Decimal: Once you pass the decimal point in the dividend, place a decimal in the quotient.
Final Step: Bring down the 25. Double the quotient (10) to get 20. Find a digit ‘$x$‘ such that $20x \times x = 1025$.
$205 \times 5 = 1025$.
Result: $\sqrt{110.25} = 10.5$
Sometimes, it is easier to convert a decimal into a fraction before solving. This allows you to work with whole numbers, which are often less confusing.
First, we rewrite the decimal as a fraction:
By the rules of square roots, we can split this:
Using the long division method on the whole number 2916:
Pairing: $\overline{29}\ \overline{16}$.
Step 1: The largest square near 29 is $5 \times 5 = 25$. Subtract 25 from 29 to get 4.
Step 2: Bring down 16 to get 416. Double the 5 to get 10.
Step 3: Find a digit to follow 10. $104 \times 4 = 416$.
So, $\sqrt{2916} = 54$.
Now, plug that back into our fraction:
Result: $\sqrt{0.2916} = 0.54$