Square root of decimals by long division

This lesson finds the square root of decimal numbers using the long division method. The key extra step is how you group the digits and where the decimal point goes in the answer.

Grouping the digits

Before doing any division, split the number into pairs of digits:

• For the part before the decimal point, group in twos from right to left.
• For the part after the decimal point, group in twos from left to right.

Then carry out the same long division as for whole-number perfect squares. When you bring down the first group from after the decimal point, place a decimal point in the quotient (and only in the quotient).

Example 1: $\sqrt{6.25}$

Group the digits as $\overline{6}.\overline{25}$.

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

$6 - 4 = 2$

Bring down $25$ to get $225$, and place the decimal point in the quotient. Double the current quotient ($2$) to get $4$ as the start of the new divisor. We need a digit $d$ with

$\overline{4d} \times d \le 225.$

Taking $d = 5$ gives $45 \times 5 = 225$, leaving remainder $0$.

$\sqrt{6.25} = 2.5$

Example 2: $\sqrt{10.24}$

Group the digits as $\overline{10}.\overline{24}$.

The largest square not exceeding $10$ is $3^2 = 9$, so the first digit is $3$.

$10 - 9 = 1$

Bring down $24$ to get $124$, and place the decimal point in the quotient. Double the quotient ($3$) to get $6$. We need a digit $d$ with

$\overline{6d} \times d \le 124.$

Taking $d = 2$ gives $62 \times 2 = 124$, leaving remainder $0$.

$\sqrt{10.24} = 3.2$

Example 3: $\sqrt{5.0625}$

Group the digits as $\overline{5}.\overline{06}\,\overline{25}$.

The largest square not exceeding $5$ is $2^2 = 4$, so the first digit is $2$.

$5 - 4 = 1$

Bring down $06$ to get $106$, and place the decimal point in the quotient. Double the quotient ($2$) to get $4$. We need a digit $d$ with

$\overline{4d} \times d \le 106.$

Taking $d = 2$ gives $42 \times 2 = 84$, and $106 - 84 = 22$.

Bring down the next group $25$ to get $2225$. Double the current quotient ($22$) to get $44$. We need a digit $d$ with

$\overline{44d} \times d \le 2225.$

Taking $d = 5$ gives $445 \times 5 = 2225$, leaving remainder $0$.

$\sqrt{5.0625} = 2.25$

Key takeaways

• Group the digits in pairs: from right to left before the decimal point, and from left to right after it.
• Place the decimal point in the quotient as soon as you bring down the first group from after the decimal point.
• All three numbers here are perfect squares: $\sqrt{6.25} = 2.5$, $\sqrt{10.24} = 3.2$, and $\sqrt{5.0625} = 2.25$.