Square Roots of Decimals by Long Division

This lesson finds the square root of numbers that are not perfect squares using the long division method, and then rounds each answer to the required number of decimal places. The key ideas are how to group the digits in pairs around the decimal point, how many pairs of zeros to add, and how to round at the end.

Setting up the problem

To find a square root by long division, group the digits in pairs. Going left from the decimal point you pair the whole-number digits, and going right from the decimal point you pair the decimal digits, adding zeros as needed to complete a pair.

The number of decimal pairs you keep controls how precisely you can round. To round to $n$ decimal places, work the answer out to $n+1$ decimal places, so you add one extra pair of zeros beyond what you need.

Example 1: $\sqrt{6.7}$ to one decimal place

To round to one decimal place, work to two decimal places, so write $6.7$ as $6.700000$ and group it: $6 \mid .70 \mid 00 \mid 00$.

The largest square not exceeding $6$ is $4$, so the first digit of the root is $2$, and $6 - 4 = 2$. Bringing down the next pair and continuing the long division gives

$\sqrt{6.7} = 2.588\ldots$

To one decimal place the second decimal digit is $8$, which is greater than $5$, so round up:

$\sqrt{6.7} \approx 2.6$

Example 2: $\sqrt{101}$ to two decimal places

Here $101$ has no decimal part, so to round to two decimal places we work to three and write $101.000000$, grouped as $1 \mid 01 \mid .00 \mid 00 \mid 00$.

The first group is $1$, a perfect square, so the root starts with $1$ and the remainder is $0$. Continuing the long division through the pairs of zeros gives

$\sqrt{101} = 10.049\ldots$

The third decimal digit is $9$, which is greater than $5$, so round the second place up:

$\sqrt{101} \approx 10.05$

Example 3: $\sqrt{8.16}$ to two decimal places

To round to two decimal places, work to three, so write $8.16$ as $8.160000$, grouped as $8 \mid .16 \mid 00 \mid 00$.

The largest square not exceeding $8$ is $4$, so the first digit of the root is $2$, and $8 - 4 = 4$. Carrying the long division through the remaining pairs gives

$\sqrt{8.16} = 2.856\ldots$

The third decimal digit is $6$, which is greater than $5$, so round the second place up:

$\sqrt{8.16} \approx 2.86$

Key takeaways

• Group the digits in pairs from the decimal point, and add one extra pair of zeros beyond the precision you need so you can round.
• Apply the long division method one pair at a time, placing the decimal point in the answer as you cross it.
• To round to two decimal places, find three decimal places: if the third digit is $5$ or more, add $1$ to the second place; if it is less than $5$, keep the first two places as they are.