Indices and surds revisited
Objective. Manipulate rational indices and surds in multi-stage exact arguments.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Surds are irrational roots such as √2 that are left in exact root form rather than rounded as decimals, and rational indices like a^(1/n) mean the nth root of a. Rules such as aᵐ×aⁿ=aᵐ⁺ⁿ and (aᵐ)ⁿ=aᵐⁿ extend to fractional and negative powers, so 8^(2/3) means (∛8)²=4. Surds are simplified by extracting square factors, e.g. √50=√(25×2)=5√2, and rationalised by multiplying by a conjugate, e.g. 1/(1+√2) becomes √2−1 after multiplying top and bottom by (√2−1). Keep exact values through multi-stage working rather than converting to decimals early.
Worked example
Simplify the surd expression √48 − √12, giving your answer in the form k√3.
- 1Write √48 = √(16×3) = 4√3
- 2Write √12 = √(4×3) = 2√3
- 3Subtract: 4√3 − 2√3
- 4Combine like surd terms to get 2√3
Answer: 2√3
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Start practisingCommon mistake. Students simplify √a+√b as √(a+b), writing √9+√16=√25=5 instead of the correct 3+4=7; surds only combine under multiplication or division, never by adding inside one root.
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