Find square root of 7056 by long division method

Question

Find the square root of 7056 using the long division method.

Solution — Step by Step

Write 7056 and group from right: 70 | 56

We work with pairs because $\sqrt{10000} = 100$ (a 2-digit number needs pairs of 4 digits, etc.).

First group: 70

Find the largest integer $n$ such that $n^2 \leq 70$ .

$8^2 = 64 \leq 70$ ✓ and $9^2 = 81 > 70$ ✗

So we write 8 as the first digit of our answer. Subtract: $70 - 64 = 6$ .

Bring down the next pair: remainder becomes 656.

The trial divisor = $2 \times \text{current quotient} = 2 \times 8 = 16$ .

We need to find digit $d$ such that $(160 + d) \times d \leq 656$ .

Try $d = 4$ : $(160 + 4) \times 4 = 164 \times 4 = 656$ ✓ (exactly equal!)

Write 4 as the next digit of the answer.

Subtract: $656 - 656 = 0$ .

The remainder is 0, so the division is exact.

\sqrt{7056} = \mathbf{84}

Verify: $84^2 = 84 \times 84 = 7056$ ✓

Why This Works

The long division method for square roots uses the algebraic identity $(a + b)^2 = a^2 + 2ab + b^2$ .

When we find the first digit (8, representing 80) and subtract, the remainder contains the $2ab + b^2$ part. The trial divisor $2 \times 80 = 160$ represents $2a$ , and we find $b = 4$ such that $(2a + b) \times b = 656 = 2(80)(4) + 4^2$ .

Common Mistake

The most common error is grouping digits incorrectly. Always group from the right, in pairs of two. For 7056, the grouping is 70|56, NOT 7|056. Wrong grouping changes the first group value and gives a wrong first digit. For odd-digit numbers like 529, group as 5|29 (the leftmost group has only 1 digit — that’s perfectly fine).

Before using long division, always check if the number ends in 1, 4, 5, 6, 9, or 0 — these are the only possible last digits of perfect squares. 7056 ends in 6 ✓. Also, the digit sum: $7+0+5+6 = 18$ , which is divisible by 9 (not a definitive test, but a quick sanity check).

Question

Solution — Step by Step

Why This Works

Common Mistake

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