Square Root of Decimal Numbers by Long Division

This lesson shows how to find the square root of a decimal number using the long division method. It covers how to group the digits, then works through perfect squares first and finishes with numbers that must be found to two decimal places.

Grouping the digits

Before you start, split the number into pairs of digits around the decimal point.

• For the whole number part (left of the decimal), group in twos starting from the decimal and moving left.
• For the fractional part (right of the decimal), group in twos starting from the decimal and moving right.
• If the last fractional group has only one digit, add a zero to complete the pair. To get extra decimal places in the answer, append more pairs of zeros, one pair per extra place.

The method is the ordinary long division square root, with the decimal point in the quotient placed as soon as you bring down the first group after the decimal.

Example: $\sqrt{10.24}$

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

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

$10 - 9 = 1$

Place the decimal point in the quotient and bring down $24$ to get $124$. Double the quotient so far: $2 \times 3 = 6$. Find a digit $d$ with $\overline{6d} \times d \le 124$. Here $62 \times 2 = 124$, so $d = 2$ and the remainder is $0$.

$\sqrt{10.24} = 3.2$

Example: $\sqrt{6.25}$

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

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

Bring down $25$ to get $225$. Double the quotient: $2 \times 2 = 4$. Then $45 \times 5 = 225$, so the next digit is $5$ with remainder $0$.

$\sqrt{6.25} = 2.5$

Example: $\sqrt{54.76}$

Group as $\overline{54}.\overline{76}$.

The largest square not exceeding $54$ is $49 = 7^2$, so the first digit is $7$ and $54 - 49 = 5$.

Bring down $76$ to get $576$. Double the quotient: $2 \times 7 = 14$. Then $144 \times 4 = 576$, so the next digit is $4$ with remainder $0$.

$\sqrt{54.76} = 7.4$

Example: $\sqrt{156.25}$

Group as $\overline{1}\,\overline{56}.\overline{25}$.

The first group is $1 = 1^2$, so the first digit is $1$ and $1 - 1 = 0$.

Bring down $56$. Double the quotient: $2 \times 1 = 2$. Then $22 \times 2 = 44 \le 56$, so the next digit is $2$ and $56 - 44 = 12$.

Place the decimal point and bring down $25$ to get $1225$. Double the quotient: $2 \times 12 = 24$. Then $245 \times 5 = 1225$, so the next digit is $5$ with remainder $0$.

$\sqrt{156.25} = 12.5$

Non-perfect squares: $\sqrt{83.4}$ to two decimal places

When the number is not a perfect square, add pairs of zeros to reach the required accuracy. The fractional part $4$ becomes $40$, and for two decimal places we append two more pairs of zeros, working with $83.\overline{40}\,\overline{00}\,\overline{00}$.

The largest square not exceeding $83$ is $81 = 9^2$, so the first digit is $9$ and $83 - 81 = 2$.

Place the decimal point and bring down $40$ to get $240$. Double the quotient: $2 \times 9 = 18$. Test the digit: $182 \times 2 = 364 > 240$, so try $1$: $181 \times 1 = 181 \le 240$. The next digit is $1$ and $240 - 181 = 59$.

Bring down $00$ to get $5900$. Double the quotient (using the digits $91$): $2 \times 91 = 182$. Testing digits, $1825 \times 5$ and $1824 \times 4$ both exceed $5900$, while $1823 \times 3 = 5469 \le 5900$, so the next digit is $3$.

Keeping two decimal places:

$\sqrt{83.4} \approx 9.13$

Example: $\sqrt{238.46}$ to two decimal places

Group as $\overline{2}\,\overline{38}.\overline{46}\,\overline{00}$, adding a pair of zeros for the second decimal place.

The first group is $2$; the largest square not exceeding it is $1 = 1^2$, so the first digit is $1$ and $2 - 1 = 1$.

Bring down $38$ to get $138$. Double the quotient: $2 \times 1 = 2$. Then $26 \times 6 = 156 > 138$, so try $5$: $25 \times 5 = 125 \le 138$. The next digit is $5$ and $138 - 125 = 13$.

Place the decimal point and bring down $46$ to get $1346$. Double the quotient: $2 \times 15 = 30$. Then $304 \times 4 = 1216 \le 1346$, so the next digit is $4$ and $1346 - 1216 = 130$.

Bring down $00$ to get $13000$. Double the quotient: $2 \times 154 = 308$. Then $3084 \times 4 = 12336 \le 13000$, so the next digit is $4$.

Keeping two decimal places:

$\sqrt{238.46} \approx 15.44$

Key takeaways

• Group digits in pairs outward from the decimal point: leftward for the whole part, rightward for the fractional part.
• Place the decimal point in the quotient as soon as you bring down the first group after the decimal.
• For non-perfect squares, add a pair of zeros for each extra decimal place you need in the answer.