Find the Square Root Correct to Two Decimal Places

This lesson explains how to find a square root correct to two decimal places. The idea is to use long division to compute one extra decimal place, then use that third place to round.

The rule for rounding to two decimal places

To give a square root correct to two decimal places, first find it to three decimal places by long division. Then look at the third decimal digit:

• If the third digit is $\geq 5$, add $1$ to the second decimal place and write the answer to two places.
• If the third digit is $< 5$, leave the second place unchanged and write the answer to two places.

Example 1: $\sqrt{7.23}$

Grouping. Before the decimal point, group the digits in pairs from right to left: $7$ is a single group. After the decimal point, group in pairs from left to right: $23$ is one group. To reach three decimal places, append extra pairs of zeros, giving

$7.\,23\,00\,00\,00$

Long division. The largest square not exceeding $7$ is $4 = 2^2$, so the first digit is $2$ with remainder $3$. Continuing the square root long division (doubling the running quotient at each stage to form the next divisor) gives the digits $2,\ 6,\ 8,\ 8$:

$\sqrt{7.23} = 2.688\ldots$

Rounding. The third decimal digit is $8$, which is $\geq 5$, so add $1$ to the second place:

$\sqrt{7.23} \approx 2.69$

Example 2: $\sqrt{0.097}$

Grouping. Pairing after the decimal point gives $09$, then $7$ which needs a zero to complete the pair, then more pairs of zeros:

$0.\,09\,70\,00$

Long division. The first group $09$ is a perfect square, $9 = 3^2$, giving the first digit $3$. Carrying the division through the next groups gives the digits $3,\ 1,\ 1$:

$\sqrt{0.097} = 0.311\ldots$

Rounding. The third decimal digit is $1$, which is $< 5$, so the second place stays the same:

$\sqrt{0.097} \approx 0.31$

Key takeaways

• To get a square root correct to two decimal places, compute it to three places first, then round.
• If the third decimal digit is $\geq 5$, round the second place up; if it is $< 5$, leave it.
• The same long division method works for any decimal, as long as you group the digits in pairs and add pairs of zeros to reach the place you need.