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Squares and Square Roots - Finding Square Roots by Division Method

Grade 8CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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The Division Method is a systematic algorithm used to find the square root of large numbers and non-perfect squares where prime factorization is not practical. It visually resembles long division but uses a changing divisor at each step.

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The process starts with grouping the digits into pairs using bars (periods) starting from the unit's place. For example, for the number 1562515625, the bars are placed as 1‾ 56‾ 25‾\overline{1}\ \overline{56}\ \overline{25}. The number of bars indicates the number of digits in the resulting square root.

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To find the first digit, determine the largest number whose square is less than or equal to the leftmost group. If the group is 5‾\overline{5}, the first digit is 22 because 22=4<52^2 = 4 < 5, whereas 32=9>53^2 = 9 > 5. This digit is written as both the divisor and the quotient.

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After subtracting and bringing down the next pair of digits, the next divisor is formed by doubling the current quotient and appending a digit 'xx' to it. The value of 'xx' is chosen such that (divisor with xx appended) ×x\times x is less than or equal to the current remainder.

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For decimal numbers, bars are placed on the integral part starting from the decimal point moving left, and on the fractional part starting from the decimal point moving right. If the last group in the fractional part is a single digit, a zero is appended to make it a pair (e.g., 12.512.5 becomes 12.5012.50).

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If the number is not a perfect square, the division can be continued by adding pairs of zeros after the decimal point to find the square root up to a desired number of decimal places.

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The relationship between the number of digits nn in a perfect square and the number of digits in its square root is given by n2\frac{n}{2} if nn is even, and n+12\frac{n+1}{2} if nn is odd.

📐Formulae

Number of digits in N=n2 (if n is even)\text{Number of digits in } \sqrt{N} = \frac{n}{2} \text{ (if } n \text{ is even)}

Number of digits in N=n+12 (if n is odd)\text{Number of digits in } \sqrt{N} = \frac{n+1}{2} \text{ (if } n \text{ is odd)}

New Divisor=(20×Current Quotient)+New Digit\text{New Divisor} = (20 \times \text{Current Quotient}) + \text{New Digit}

x×y=x×y\sqrt{x \times y} = \sqrt{x} \times \sqrt{y}

xy=xy\sqrt{\frac{x}{y}} = \frac{\sqrt{x}}{\sqrt{y}}

💡Examples

Problem 1:

Find the square root of 729729 using the division method.

Solution:

Step 1: Group the digits in pairs from the right: 7‾ 29‾\overline{7}\ \overline{29}. There are two groups, so the square root will have 2 digits.\Step 2: Find the largest square less than or equal to 77. That is 22=42^2 = 4. Write 22 as the divisor and quotient. Subtract 44 from 77 to get remainder 33.\Step 3: Bring down the next pair 2929. The new dividend is 329329.\Step 4: Double the quotient (2×2=42 \times 2 = 4). We need to find a digit xx such that 4x×x≤3294x \times x \le 329. Try x=7x=7: 47×7=32947 \times 7 = 329.\Step 5: Since the remainder is 00, the process ends. The quotient is 2727.\Therefore, 729=27\sqrt{729} = 27.

Explanation:

We first paired the digits and found the largest square for the first group. We then updated the divisor by doubling the existing quotient and finding the digit that completes the product.

Problem 2:

Find the square root of 17.6417.64 using the division method.

Solution:

Step 1: Place bars over the pairs: 17‾.64‾\overline{17}.\overline{64}.\Step 2: Find the largest square ≤17\le 17. 42=164^2 = 16. Divisor = 44, Quotient = 44. Remainder 17−16=117 - 16 = 1.\Step 3: Place a decimal point in the quotient after 44. Bring down the pair 6464. New dividend is 164164.\Step 4: Double the quotient (4×2=84 \times 2 = 8). Find a digit xx such that 8x×x≤1648x \times x \le 164. Try x=2x=2: 82×2=16482 \times 2 = 164.\Step 5: Remainder is 00. The quotient is 4.24.2.\Therefore, 17.64=4.2\sqrt{17.64} = 4.2.

Explanation:

For decimals, the process is identical to whole numbers, but we must ensure the decimal point is placed in the quotient immediately after we cross the decimal in the dividend.