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What is the simplified form of $\sqrt{72}$?

The simplified form of 72\sqrt{72} is 626\sqrt{2}.

To simplify 72\sqrt{72}, we begin by determining its prime factors. We can break down 7272 as follows:

72=2×36=2×2×18=2×2×2×9=2×2×2×3×3.72 = 2 \times 36 = 2 \times 2 \times 18 = 2 \times 2 \times 2 \times 9 = 2 \times 2 \times 2 \times 3 \times 3.

Next, we can group the prime factors in pairs, resulting in:

(2×2)×(3×3)×2.(2 \times 2) \times (3 \times 3) \times 2.

Each pair of identical factors can be taken out of the square root as a single number. Therefore, we have:

72=(22)×(32)×2=2×3×2=62.\sqrt{72} = \sqrt{(2^2) \times (3^2) \times 2} = 2 \times 3 \times \sqrt{2} = 6\sqrt{2}.

This indicates that the square root of 7272 can be expressed as 66 times the square root of 22.

The technique of simplifying square roots by factoring them into prime numbers and then grouping pairs is a valuable skill in GCSE Maths. This method allows for the transformation of complex square roots into simpler forms, making them easier to handle in subsequent calculations.

Answered by: Prof. Richard White
A-Level Maths Tutor
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