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

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

To express 72\sqrt{72} in its simplest surd form, we start by factorizing 7272 into its prime factors. The prime factorization of 7272 is given by 2×2×2×3×32 \times 2 \times 2 \times 3 \times 3, which can be represented as 23×322^3 \times 3^2. To simplify the square root, we need to identify pairs of prime factors. Each pair of identical numbers can be taken out of the square root as a single number.

In this case, we have one pair of 22s, represented as 222^2, and one pair of 33s, represented as 323^2. When we extract these pairs from the square root, we multiply them together: 2×32 \times 3, which equals 66. The remaining factor inside the square root is 22, since we have one 22 left over from the factorization. Consequently, we find that 72\sqrt{72} simplifies to 626\sqrt{2}.

This process illustrates that surds provide a method for expressing irrational numbers in their simplest form. By breaking down the number into its prime factors and simplifying, we make calculations more manageable. The objective is to express the number in a way that the square root is as simplified as possible, while retaining any irrational components within the square root.

Answered by: Prof. Michael Lewis
IB Physics Tutor
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