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Surds & Indices

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Quantitative Aptitude - Surds & Indices

Master radical simplification and exponent rules to simplify expressions swiftly.

1. Surds (Radicals)

  • Surd form: \( \sqrt{n} \), \( \sqrt[3]{n} \), etc., when \( n \) is not a perfect power.
  • Simplify by prime factorization under the root (e.g., \( \sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2} \)).
  • Rationalize denominators to remove radicals.

2. Indices (Exponents)

  • Basic rules: \( a^m \times a^n = a^{m+n};\; a^m ÷ a^n = a^{m−n};\; (a^m)^n = a^{mn} \).
  • Negative exponent: \( a^{−n} = 1/a^n \).
  • Fractional exponent: \( a^{m/n} = \sqrt[n]{a^m} \).

3. Combined Expressions

  • Combine like bases using exponent laws.
  • Convert surd form into exponent form to simplify.

4. Quick Tips

  • Always simplify radical part under the root first.
  • Cancel exponents immediately when bases match.
  • Rationalize by multiplying top and bottom by conjugate if needed.

5. Common Mistakes

  • Incorrect exponent rules application.
  • Forgetting to rationalize denominators in final form.

6. Revision Checklist

  • Practice surd simplification and exponent laws weekly.
  • Memorize prime factor pairs for root extraction.

Summary: Simplify surds by extracting squares, apply exponent rules cleanly, rationalize denominators, and use exponent-surd conversions for fast and precise solutions.

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