Overview
This lecture introduces radicals (roots), explains how they relate to exponents, demonstrates radical notation, and discusses key properties and calculation rules for roots.
Exponents and Bases
- An exponent shows how many times a base is multiplied by itself (e.g., 2³ = 2 × 2 × 2 = 8).
- The base is the number being raised to a power; the exponent tells how many times it is used.
Introduction to Radicals (Roots)
- A radical or root finds the base given the exponent and result (e.g., the square root of 4 is 2).
- The radical symbol √ represents roots; the number under the symbol is the radicand.
- The small number above the radical symbol (index) shows which root (e.g., ³√ for cube root).
- For square roots, the index 2 is often omitted (√9 instead of ²√9).
Calculating Roots
- To find the nth root of a number, rewrite the relation as "the nth root of y = x" if xⁿ = y.
- Use calculators with root or radical functions for non-square roots; input may vary by device.
Properties and Special Cases
- The radicand is the number under the radical sign.
- Square roots of positive numbers have two answers: positive and negative (±), because both squared give the radicand.
- Even-index roots (n=2, 4, 6, ...) of positive numbers have both positive and negative solutions.
- Odd-index roots (n=3, 5, 7, ...) of positive numbers yield a single positive answer; of negative numbers, a single negative answer.
- Even roots of negative numbers do not yield real number answers (at this level).
Example Problems
- 5th root of 243 is 3 (since 3⁵ = 243); only positive answer because the index is odd.
- 6th root of 117,649 is ±7 (since both 7⁶ and (-7)⁶ = 117,649); both answers since the index is even.
- 4th root of 6.5536 is ±1.6; applies to decimals as well.
- 3rd root of -10.648 is -2.2 (odd root of a negative number gives a negative answer).
Key Terms & Definitions
- Exponent — the number that tells how many times to multiply the base by itself.
- Base — the number being raised to a power by the exponent.
- Radical (Root) — the mathematical operation that finds the base given the result and exponent.
- Radicand — the number under the radical symbol to be rooted.
- Index (Order) — the small number showing which root to take (e.g., ³√ for cube root).
- Square Root — the root with index 2, usually written without the 2.
Action Items / Next Steps
- Practice finding roots with and without a calculator.
- Review simplification of roots using prime factorization (see upcoming video/playlist).
- Remember: For real answers, do not take even roots of negative radicands at this level.