Summary
Algebraic roots and indices involve applying index notation to simplify algebraic expressions using the laws of indices. These laws help in representing products efficiently, especially when dealing with multiple terms.
- Index Notation — a way to represent repeated multiplication of a number or variable. Example: 3^4 means 3 multiplied by itself 4 times.
- Product of Powers Rule — when multiplying like bases, add the exponents. Example: a^m x a^n = a^(m+n), like 3^2 x 3^4 = 3^(2+4) = 3^6.
- Quotient of Powers Rule — when dividing like bases, subtract the exponents. Example: a^m ÷ a^n = a^(m-n), like 5^9 ÷ 5^3 = 5^(9-3) = 5^6.
- Power of a Power Rule — when raising a power to another power, multiply the exponents. Example: (a^m)^n = a^(mn), like (4^2)^5 = 4^(25) = 4^10.
- Negative Exponent Rule — a negative exponent means the reciprocal of the base raised to the positive exponent. Example: a^-m = 1/a^m, like 9^-3 = 1/9^3.
- Fractional Exponent Rule — a fractional exponent represents a root. Example: a^(m/n) = the n-th root of a raised to the power of m, like 16^(1/2) = √16 = 4.
- Zero Exponent Rule — any base raised to the power of zero is 1. Example: a^0 = 1.
Exam Tips
Key Definitions to Remember
- Index Notation
- Product of Powers Rule
- Quotient of Powers Rule
- Power of a Power Rule
- Negative Exponent Rule
- Fractional Exponent Rule
- Zero Exponent Rule
Common Confusions
- Confusing the product and quotient rules.
- Misapplying the negative exponent rule as a subtraction.
- Forgetting that any number to the power of zero is 1.
Typical Exam Questions
- Simplify 3^2 x 3^4? Answer: 3^6
- What is 5^9 ÷ 5^3? Answer: 5^6
- Simplify (4^2)^5? Answer: 4^10
- What is 9^-3? Answer: 1/9^3
- Simplify 16^(1/2)? Answer: 4
What Examiners Usually Test
- Ability to apply the laws of indices correctly.
- Simplifying expressions using index notation.
- Understanding and using fractional and negative indices.
