Laws of Indices

What are the laws of indices?

Laws of indices (or index laws) allow you to simplify and manipulate expressions involving exponents
- An exponent is a power that a number (called the base) is raised to
- Laws of indices can be used when the numbers are written with the same base
The index laws you need to know are:
- ${(x y)}^{m} = x^{m} y^{m}$
- ${(\frac{x}{y})}^{m} = \frac{x^{m}}{y^{m}}$
- $x^{m} \times x^{n} = x^{m + n}$
- $x^{m} \div x^{n} = x^{m - n}$
- ${(x^{m})}^{n} = x^{m n}$
- $x^{1} = x$
- $x^{0} = 1$
- $\frac{1}{x^{m}} = x^{- m}$
- $x^{\frac{1}{n}} = \sqrt[n]{x}$
- $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$
These laws are not in the formula booklet so you must remember them

How are laws of indices used?

You will need to be able to carry out multiple calculations with the laws of indices
- Take your time and apply each law individually
- Work with numbers first and then with algebra
Index laws only work with terms that have the same base, make sure you change the base of the term before using any of the index laws
- Changing the base means rewriting the number as an exponent with the base you need
- For example, $9^{4} = (3^{2})^{4} = 3^{2 \times 4} = 3^{8}$
- Using the above can them help with problems like $9^{4} \div 3^{7} = 3^{8} \div 3^{7} = 3^{1} = 3$

Examiner Tip

Index laws are rarely a question on their own in the exam but are often needed to help you solve other problems, especially when working with logarithms or polynomials
Look out for times when the laws of indices can be applied to help you solve a problem algebraically

Worked example

Simplify the following equations:

\frac{(3 x^{2}) (2 x^{3} y^{2})}{(6 x^{2} y)}

ai-sl-1-1-2-laws-of-indices-we-i

ii)

(4 x^{2} y^{- 4})^{3} (2 x^{3} y^{- 1})^{- 2}

Laws of Indices (DP IB Maths: AA HL)

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What are the laws of indices?

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1. Number & Algebra

1.1 Number & Algebra Toolkit

1.1.1 Standard Form

1.1.2 Laws of Indices

1.1.3 Partial Fractions

1.2 Exponentials & Logs

1.2.1 Introduction to Logarithms

1.2.2 Laws of Logarithms

1.2.3 Solving Exponential Equations

1.3 Sequences & Series

1.3.1 Language of Sequences & Series

1.3.2 Arithmetic Sequences & Series

1.3.3 Geometric Sequences & Series

1.3.4 Applications of Sequences & Series

1.3.5 Compound Interest & Depreciation

1.4 Simple Proof & Reasoning

1.4.1 Proof

1.5 Further Proof & Reasoning

1.5.1 Proof by Induction

1.5.2 Proof by Contradiction

1.6 Binomial Theorem

1.6.1 Binomial Theorem

1.6.2 Extension of The Binomial Theorem

1.7 Permutations & Combinations

1.7.1 Counting Principles

1.7.2 Permutations & Combinations

1.8 Complex Numbers

1.8.1 Intro to Complex Numbers

1.8.2 Modulus & Argument

1.8.3 Introduction to Argand Diagrams

1.9 Further Complex Numbers

1.9.1 Geometry of Complex Numbers

1.9.2 Forms of Complex Numbers

1.9.3 Complex Roots of Polynomials

1.9.4 De Moivre's Theorem

1.9.5 Roots of Complex Numbers

1.10 Systems of Linear Equations

1.10.1 Systems of Linear Equations

1.10.2 Algebraic Solutions

2. Functions

2.1 Linear Functions & Graphs

2.1.1 Equations of a Straight Line

2.2 Quadratic Functions & Graphs

2.2.1 Quadratic Functions

2.2.2 Factorising & Completing the Square

2.2.3 Solving Quadratics

2.2.4 Quadratic Inequalities

2.2.5 Discriminants

2.3 Functions Toolkit

2.3.1 Language of Functions

2.3.2 Composite & Inverse Functions

2.3.3 Symmetry of Functions

2.3.4 Graphing Functions

2.4 Other Functions & Graphs

2.4.1 Exponential & Logarithmic Functions

2.4.2 Solving Equations

2.4.3 Modelling with Functions

2.5 Reciprocal & Rational Functions

2.5.1 Reciprocal & Rational Functions

2.6 Transformations of Graphs

2.6.1 Translations of Graphs

2.6.2 Reflections of Graphs

2.6.3 Stretches Graphs

2.6.4 Composite Transformations of Graphs

2.7 Polynomial Functions

2.7.1 Factor & Remainder Theorem

2.7.2 Polynomial Division

2.7.3 Polynomial Functions

2.7.4 Roots of Polynomials

2.8 Inequalities

2.8.1 Solving Inequalities Graphically

2.8.2 Polynomial Inequalities

2.9 Further Functions & Graphs