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What is the surd form of $\sqrt{48}$?

The surd form of 48\sqrt{48} can be expressed as 434\sqrt{3}.

To simplify 48\sqrt{48} into its simplest surd form, we begin by factorizing the number 4848 into its prime factors:

48=2×24=2×2×12=2×2×2×6=2×2×2×2×348 = 2 \times 24 \\ = 2 \times 2 \times 12 \\ = 2 \times 2 \times 2 \times 6 \\ = 2 \times 2 \times 2 \times 2 \times 3

Next, we can group these factors into pairs of the same number:

48=(2×2)×(2×2)×3=4×4×348 = (2 \times 2) \times (2 \times 2) \times 3 \\ = 4 \times 4 \times 3

Using the property that the square root of a product equals the product of the square roots, we can simplify further:

48=(4×4×3)=(4×4)×3=43\sqrt{48} = \sqrt{(4 \times 4 \times 3)} \\ = \sqrt{(4 \times 4)} \times \sqrt{3} \\ = 4\sqrt{3}

Therefore, the surd form of 48\sqrt{48} is 434\sqrt{3}. This indicates that 48\sqrt{48} can be simplified to 44 times the square root of 33. This surd form is advantageous because it is simpler and more manageable for subsequent calculations.

Answered by: Dr. Emily Clark
GCSE Maths Tutor
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