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Squares and Square Roots - Finding Square Roots by Prime Factorisation

Grade 8ICSE

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

🔑Concepts

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The square root of a number xx is a value that, when multiplied by itself, gives xx, denoted as x\sqrt{x}. Visually, if you represent a perfect square number as the area of a square shape, the square root represents the length of one side of that square. For example, a square with an area of 1616 units is made of 4×44 \times 4 units, so 16=4\sqrt{16} = 4.

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Prime Factorisation is the process of breaking a number down into its basic building blocks of prime numbers, such as 2,3,5,7,…2, 3, 5, 7, \dots. This can be visualized as a 'Factor Tree', where the number branches out into smaller and smaller factors until only prime numbers remain at the ends of all branches.

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A number is a perfect square only if every prime factor in its factorisation can be grouped into identical pairs. Visually, if you were to arrange the prime factors into sets, every factor must have a 'partner' of the same value. If any factor is left alone, the number is not a perfect square.

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To calculate the square root using prime factorisation, we first list all prime factors, group them into pairs of two identical primes, and then select exactly one representative from each pair. The product of these selected representatives is the square root. For example, if a number nn factors into 2×2×3×32 \times 2 \times 3 \times 3, its square root is 2×3=62 \times 3 = 6.

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Squaring and finding square roots are inverse operations. This relationship can be visualized as a two-way process: squaring moves from a side length to a total area (side2=areaside^2 = area), while finding the square root moves from the total area back to the side length (area=side\sqrt{area} = side).

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Finding the smallest multiplier or divisor to create a perfect square involves identifying 'unpaired' prime factors. To make a number a perfect square, you must either multiply by the unpaired factor (to complete the pair) or divide by it (to remove the incomplete pair). Visually, this is like adding or removing blocks to ensure every block has a matching partner.

📐Formulae

If x2=yx^2 = y, then y=x\sqrt{y} = x

a×b=a×b\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}

p12×p22×⋯×pn2=p1×p2×⋯×pn\sqrt{p_1^2 \times p_2^2 \times \dots \times p_n^2} = p_1 \times p_2 \times \dots \times p_n

💡Examples

Problem 1:

Find the square root of 1102511025 using the prime factorisation method.

Solution:

Step 1: Perform prime factorisation of 1102511025. \ 11025=3×3×5×5×7×711025 = 3 \times 3 \times 5 \times 5 \times 7 \times 7 \ Step 2: Group the factors into identical pairs. \ 11025=(3×3)×(5×5)×(7×7)11025 = (3 \times 3) \times (5 \times 5) \times (7 \times 7) \ Step 3: Pick one factor from each pair. \ The factors are 33, 55, and 77. \ Step 4: Multiply these factors to find the square root. \ 11025=3×5×7=105\sqrt{11025} = 3 \times 5 \times 7 = 105

Explanation:

By breaking the number into its prime components, we identify that it is composed of pairs of 33, 55, and 77. Taking one from each pair and multiplying them gives us the side length of the square, which is 105105.

Problem 2:

Find the smallest number by which 392392 must be divided so that the quotient is a perfect square. Also, find the square root of that quotient.

Solution:

Step 1: Find the prime factorisation of 392392. \ 392=2×2×2×7×7392 = 2 \times 2 \times 2 \times 7 \times 7 \ Step 2: Group the prime factors into pairs. \ 392=(2×2)×2×(7×7)392 = (2 \times 2) \times 2 \times (7 \times 7) \ Step 3: Identify the unpaired factor. \ The factor 22 does not have a pair. \ Step 4: Divide the original number by this unpaired factor. \ 392÷2=196392 \div 2 = 196 \ Step 5: Find the square root of the resulting perfect square. \ Prime factors of 196196 are (2×2)×(7×7)(2 \times 2) \times (7 \times 7). \ 196=2×7=14\sqrt{196} = 2 \times 7 = 14

Explanation:

In the factorisation of 392392, the prime factor 22 is left without a partner. Dividing the number by 22 removes this 'extra' factor, leaving only complete pairs, which results in the perfect square 196196.