Square Root of 6.27 and 123.8 to Decimal Places

This lesson finds the square root of two decimal numbers that are not perfect squares, using the long division method. The key ideas are grouping the digits around the decimal point and adding pairs of zeros to reach the number of decimal places we want.

Grouping a decimal for long division

To take a square root by long division, group the digits in pairs. Before the decimal point, group from the right; after the decimal point, group from the left. Each extra pair of zeros after the decimal point gives one more decimal place in the answer, so to find a root to three places we add three pairs of zeros, and to two places we add two.

Square root of 6.27 to three decimal places

We want $\sqrt{6.27}$ to three decimal places, so we write it as

$6.\,27\;00\;00\;00$

The first group is $6$. The largest perfect square not exceeding $6$ is $4 = 2^2$, so the first digit of the root is $2$.

$6 - 4 = 2$

Bring down 27. Double the quotient so far: $2 \times 2 = 4$. We need a digit $d$ so that $\overline{4d}\times d \le 227$. Trying $d = 5$ gives $45 \times 5 = 225 \le 227$, so the next digit is $5$.

$227 - 225 = 2$

Bring down 00. Double the quotient $25$ to get $50$. We need $\overline{50d}\times d \le 200$. Even $d = 1$ gives $501 \times 1 = 501 > 200$, so the digit is $0$.

$200 - 0 = 200$

Bring down 00. Double the quotient $250$ to get $500$. We need $\overline{500d}\times d \le 20000$. Trying $d = 4$ gives $5004 \times 4 = 20016 > 20000$, so take $d = 3$: $5003 \times 3 = 15009 \le 20000$. The next digit is $3$.

Reading the quotient with the decimal point in place,

$\sqrt{6.27} \approx 2.503$

Square root of 123.8 to two decimal places

We want $\sqrt{123.8}$ to two decimal places, so we write it as

$1\;23.\,80\;00$

The first group is $1$, a perfect square with root $1$, so the first digit is $1$.

$1 - 1 = 0$

Bring down 23. Double the quotient: $2 \times 1 = 2$. We need $\overline{2d}\times d \le 23$. Trying $d = 1$ gives $21 \times 1 = 21 \le 23$, so the next digit is $1$.

$23 - 21 = 2$

Bring down 80. Double the quotient $11$ to get $22$. We need $\overline{22d}\times d \le 280$. Trying $d = 1$ gives $221 \times 1 = 221 \le 280$ (while $d = 2$ gives $444 > 280$), so the next digit is $1$.

$280 - 221 = 59$

Bring down 00. Double the quotient $111$ to get $222$. We need $\overline{222d}\times d \le 5900$. Trying $d = 2$ gives $2222 \times 2 = 4444 \le 5900$, so the next digit is $2$.

Reading the quotient with the decimal point in place,

$\sqrt{123.8} \approx 11.12$

If instead we wanted the answer correct to two decimal places, we would find one more digit and round: if it is $5$ or more, add $1$ to the second decimal place.

Key takeaways

• Group the digits of a decimal in pairs, from the right before the point and from the left after it.
• Add one pair of zeros for each decimal place of accuracy you want in the root.
• At each step, double the quotient so far and find the largest digit $d$ with $\overline{(\text{double})d}\times d$ not exceeding the current number.
• $\sqrt{6.27} \approx 2.503$ and $\sqrt{123.8} \approx 11.12$.