Square Root of Perfect Squares by Long Division

This lesson finds the square root of perfect squares using the long division method, building up from a three digit number to a six digit number. The routine is always the same: group the digits in pairs from the right, work group by group, and double the running answer to make each new divisor.

The setup

To prepare a number, group its digits into pairs starting from the right. The leftmost group may have one or two digits. You should know the perfect squares up to $10^2 = 100$ by heart, since at each step you compare a group against the nearest perfect square at or below it.

Square root of 841

Group as $8\;|\;41$.

Left group

The largest square not exceeding $8$ is $4 = 2^2$, so the first quotient digit is $2$. Place $2$ as both the divisor and the quotient:

$8 - 2\times 2 = 8 - 4 = 4.$

Bring down 41

The remainder $4$ with the next group gives $441$. Double the quotient: $2\times 2 = 4$, and leave one place for the next digit, so the new divisor is $4\_$. The number ends in $1$, and only $1$ or $9$ squared ends in $1$. Testing $9$:

$49 \times 9 = 441,$

which matches exactly. So the next quotient digit is $9$ and the remainder is $0$.

$\sqrt{841} = 29.$

Square root of 5776

Group as $57\;|\;76$.

Left group

The largest square not exceeding $57$ is $49 = 7^2$, so the first digit is $7$.

$57 - 49 = 8.$

Bring down 76

This gives $876$. Double the quotient: $7\times 2 = 14$, so the new divisor is $14\_$. Testing the last digit, $146 \times 6 = 876$, which matches.

$\sqrt{5776} = 76.$

Square root of 10816

Group as $1\;|\;08\;|\;16$.

Left group

$1 = 1^2$, so the first digit is $1$ and the remainder is $0$.

Bring down 08, a zero appears

The group is $08$. Double the quotient: $1\times 2 = 2$, so the divisor starts at $2\_$. Since $8$ cannot be divided by $20$, the quotient digit here is $0$. The running divisor becomes $20$.

Bring down 16

This gives $816$. Testing the last digit, $204 \times 4 = 816$, which matches.

$\sqrt{10816} = 104.$

Square root of 106276

Group as $10\;|\;62\;|\;76$.

Left group

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

$10 - 9 = 1.$

Bring down 62

This gives $162$. Double the quotient: $3\times 2 = 6$, so the divisor is $6\_$. Testing $62 \times 2 = 124 \le 162$, so the next digit is $2$ and the remainder is $162 - 124 = 38$.

Bring down 76

This gives $3876$. Double the running quotient $32$: $32\times 2 = 64$, so the divisor is $64\_$. Testing the last digit, $646 \times 6 = 3876$, which matches.

$\sqrt{106276} = 326.$

Key takeaways

• Group the digits in pairs from the right, then work one group at a time from the left.
• For each group, choose the quotient digit by comparing against the largest perfect square at or below it, then double the running quotient to form the next divisor.
• If no digit fits a group, the quotient digit is $0$, as with the four digit example.