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Squares and Square Roots - Finding Square Roots by Prime Factorisation

Grade 8CBSE

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

🔑Concepts

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A square root is the inverse operation of squaring a number. If a2=ba^2 = b, then b=a\sqrt{b} = a. Visually, if you have a square area of bb units, the length of one side is b\sqrt{b}.

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Prime Factorization involves breaking down a composite number into a product of prime numbers. This can be visualized as a 'Factor Tree' where the number at the top branches down until every branch ends in a prime factor.

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For a number to be a perfect square, every prime factor in its prime factorization must occur an even number of times, allowing them to be grouped into identical pairs.

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The process of finding the square root involves three main visual/logical steps: 1. Prime factorize the number, 2. Group the identical factors into pairs, 3. Take one factor from each pair and multiply them.

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If any prime factor remains single (unpaired) after grouping, the number is not a perfect square. To make it a perfect square, you must either multiply or divide the number by that specific unpaired factor.

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The square root symbol \sqrt{\quad} represents the positive square root of a number. For example, 25=5\sqrt{25} = 5 because 5×5=255 \times 5 = 25.

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The property x2=x\sqrt{x^2} = x is the fundamental rule used in this method. By expressing a number as (p1×p2×...)2(p_1 \times p_2 \times ...)^2, we can easily identify the root.

📐Formulae

x2=x\sqrt{x^2} = x

If N=p1×p1×p2×p2×...N = p_1 \times p_1 \times p_2 \times p_2 \times ..., then N=p1×p2×...\sqrt{N} = p_1 \times p_2 \times ...

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

Area of a Square = side×sideside \times side, therefore side=Areaside = \sqrt{Area}

💡Examples

Problem 1:

Find the square root of 17641764 using the prime factorization method.

Solution:

Step 1: Perform prime factorization of 17641764. 1764=2×8821764 = 2 \times 882 882=2×441882 = 2 \times 441 441=3×147441 = 3 \times 147 147=3×49147 = 3 \times 49 49=7×749 = 7 \times 7 So, 1764=2×2×3×3×7×71764 = 2 \times 2 \times 3 \times 3 \times 7 \times 7.

Step 2: Group the factors into pairs. 1764=(2×2)×(3×3)×(7×7)1764 = (2 \times 2) \times (3 \times 3) \times (7 \times 7)

Step 3: Take one factor from each pair. 1764=2×3×7=42\sqrt{1764} = 2 \times 3 \times 7 = 42.

Explanation:

We break the number down into its smallest prime components. Since every prime factor (2,3,2, 3, and 77) appears as a pair, we take one representative from each pair and multiply them to find the root.

Problem 2:

Find the smallest number by which 252252 must be multiplied to get a perfect square. Also, find the square root of the perfect square so obtained.

Solution:

Step 1: Prime factorize 252252. 252=2×2×3×3×7252 = 2 \times 2 \times 3 \times 3 \times 7

Step 2: Identify the unpaired factor. The prime factors 22 and 33 are in pairs, but 77 is alone. To make it a pair, we must multiply 252252 by 77. New number = 252×7=1764252 \times 7 = 1764.

Step 3: Find the square root of the new number. 1764=(2×2)×(3×3)×(7×7)1764 = (2 \times 2) \times (3 \times 3) \times (7 \times 7) 1764=2×3×7=42\sqrt{1764} = 2 \times 3 \times 7 = 42.

Explanation:

In prime factorization, any factor without a partner prevents the number from being a perfect square. By multiplying by that missing factor (77), we complete the pair. The square root is then found by taking one number from each completed pair.