144 = 2⁴ × 3² → √144 = 2² × 3 = 12.

Find Square Root of 144 by Prime Factorisation

Question

Find the square root of 144 using the prime factorisation method.

Solution — Step by Step

We break 144 into its prime factors by dividing repeatedly by the smallest prime.

144 \div 2 = 72 \div 2 = 36 \div 2 = 18 \div 2 = 9 \div 3 = 3 \div 3 = 1

So: $144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3$

Group the repeated factors using exponents — this is where the method becomes clean.

144 = 2^4 \times 3^2

The rule is simple: $\sqrt{a^m} = a^{m/2}$ . We halve each exponent.

\sqrt{144} = \sqrt{2^4 \times 3^2} = 2^{4/2} \times 3^{2/2} = 2^2 \times 3^1

Now we just multiply.

2^2 \times 3 = 4 \times 3 = \mathbf{12}

So $\sqrt{144} = \mathbf{12}$ .

Why This Works

Every perfect square has prime factors that appear an even number of times. That is not a coincidence — it is the definition of a perfect square. When we take the square root, we are asking: “what number, multiplied by itself, gives 144?” Multiplying by itself doubles all the exponents, so square-rooting halves them.

For $144 = 2^4 \times 3^2$ : the exponents are 4 and 2 — both even. This confirms 144 is a perfect square before we even finish the calculation. If any exponent had been odd (say, $2^3 \times 3^2 = 72$ ), 72 would not be a perfect square.

This method scales to any number. You do not need to guess or remember. Factor it, check if all exponents are even, halve them, multiply out.

Alternative Method — Repeated Subtraction

There is a pattern-based method for small perfect squares: subtract consecutive odd numbers from 144 until you reach zero. The count of subtractions is the square root.

144 - 1 = 143 - 3 = 140 - 5 = 135 - 7 = 128 - 9 = 119 - 11 = 108 - 13 = 95

95 - 15 = 80 - 17 = 63 - 19 = 44 - 21 = 23 - 23 = 0

We subtracted 12 odd numbers, so $\sqrt{144} = 12$ .

Prime factorisation is faster for NCERT problems and board exams. Use repeated subtraction only to verify small answers quickly — it gets tedious beyond 100.

Common Mistake

Students often pair factors incorrectly. They write $144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3$ and then take one factor from each pair to get $2 \times 2 \times 3 = 12$ — which is actually correct here, but they do not understand why they are picking one from each pair.

The rule is: make pairs of identical factors, then take one factor from each pair. If any factor is left unpaired, the number is not a perfect square. For 144 we get pairs $(2,2), (2,2), (3,3)$ — three pairs, giving us $2 \times 2 \times 3 = 12$ . Understanding the pairing logic protects you when questions ask you to find the smallest number by which to multiply or divide to make something a perfect square.

Question

Solution — Step by Step

Why This Works

Alternative Method — Repeated Subtraction

Common Mistake

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