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Surd

Surd

An irrational number that can be expressed as a radical, such as The square root of 5 or The fifth root of 7, written as a radical expression with index 5 and radicand 7..

Worked Example

Problem: Determine which of the following are surds: 5\sqrt{5}, 16\sqrt{16}, 83\sqrt[3]{8}, 12\sqrt{12}.
Step 1: Check 5\sqrt{5}. Since 5 is not a perfect square, this radical cannot be simplified to a rational number.
5≈2.236…(irrational — this is a surd)\sqrt{5} \approx 2.236\ldots \quad \text{(irrational — this is a surd)}
Step 2: Check 16\sqrt{16}. Since 4×4=164 \times 4 = 16, this simplifies to a whole number.
16=4(rational — not a surd)\sqrt{16} = 4 \quad \text{(rational — not a surd)}
Step 3: Check 83\sqrt[3]{8}. Since 23=82^3 = 8, this simplifies to a whole number.
83=2(rational — not a surd)\sqrt[3]{8} = 2 \quad \text{(rational — not a surd)}
Step 4: Check 12\sqrt{12}. You can simplify 12=23\sqrt{12} = 2\sqrt{3}, but 3\sqrt{3} is still irrational, so the expression remains irrational.
12=23≈3.464…(irrational — this is a surd)\sqrt{12} = 2\sqrt{3} \approx 3.464\ldots \quad \text{(irrational — this is a surd)}
Answer: 5\sqrt{5} and 12\sqrt{12} are surds. 16\sqrt{16} and 83\sqrt[3]{8} are not surds.

Why It Matters

Surds let you write exact irrational values instead of rounding decimals. When solving equations like x2=3x^2 = 3, the exact answer x=3x = \sqrt{3} is a surd, and keeping it in that form avoids rounding errors in further calculations. Simplifying and manipulating surds is a core skill in algebra and geometry, especially when working with the Pythagorean theorem or the quadratic formula.

Common Mistakes

Mistake: Calling every square root a surd, including roots like 25\sqrt{25} or 49\sqrt{49}.
Correction: A radical is only a surd if it cannot be simplified to a rational number. 25=5\sqrt{25} = 5, which is rational, so it is not a surd.

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