Lecture Notes: Indices and Index Laws at National 5 Level

Introduction

• Focus on understanding indices and index laws.
• Covered each past paper in-depth.
• Indices are essentially powers.

Laws of Indices

Law 1: Multiplying Powers

• Formula: ( a^m \times a^n = a^{m+n} ).
• Example: ( a^3 \times a^2 = a^5 ).
• Important: The bases must be the same (e.g., cannot apply to ( a^3 \times b^2 )).

Law 2: Dividing Powers

• Formula: ( a^m \div a^n = a^{m-n} ).
• Applies to fractions as division.

Power of a Power

• Formula: ( (a^m)^n = a^{m \times n} ).
• Example: ( (3^2)^3 = 3^6 ).
• Be careful when combining numbers and letters:
  • Example: ( (3a^2)^4 = 3^4 \times a^{8} ).

Zero Power

• Formula: ( a^0 = 1 ).
• Applies to any number or variable.

Negative Indices

• ( a^{-m} = \frac{1}{a^m} ).
• Conversely: ( \frac{1}{a^m} = a^{-m} ).
• Answers often need to be left as positive powers.

Fractional Powers

• ( a^{\frac{1}{n}} = \sqrt[n]{a} ).
• Example: ( a^{\frac{1}{2}} = \sqrt{a} ).

General Fractional Powers

• ( a^{\frac{m}{n}} = \sqrt[n]{a^m} ).
• Example: ( a^{\frac{3}{4}} = \sqrt[4]{a^3} ).

Past Paper Questions Examples

2014 Paper 2 Question 8

• Simplification using laws of indices.

2015 Paper 1 Question 14

• Evaluating (8^{\frac{5}{3}}).
• Cube root approach simplifies calculation.

2016 Paper 2 Question 10

• Simplifying expressions with negative indices to positive.

2017 Paper 2 Question 12

• Rewriting (\frac{1}{\sqrt[3]{x}}) using indices.

2018 Paper 1 Question 15

• Removing brackets and simplifying ((\frac{2}{3}p^{4})^2).

2019 Paper 2 Question 16

• Expression simplification involving square roots.

More Examples

• Evaluation of fractional powers such as (16^{3/2}).
• Simplification like (m^{-2}) to positive indices.

Conclusion

• Understanding and application of index laws are key.
• Practice with past paper questions enhances skills.
• Regular review of indices laws aids in mastery.