Bluebird Notes
Grade 11 Mathematics
Chapter 1
Exponents and Surds
2026 Edition
Complete Chapter Notes for the IEB Curriculum
Comprehensive Theory • Worked Examples • Exam Tips
1
© 2026 Bluebird Notes. All rights reserved.
,Copyright
© 2026 Bluebird Notes
All rights reserved.
No part of this publication may be reproduced, distributed or shared in any form without prior
written permission from Bluebird Notes.
These notes are intended for personal educational use only.
Contact
Bluebird Notes
© Bluebird Notes
2
© 2026 Bluebird Notes. All rights reserved.
,Table of Contents
Chapter 1: page 4
1.1 What is an Exponent: page 4
1.2 Laws of Exponents: page 6
1.3 Numerical Bases: page 6
1.4 Expressions with Irrational Exponents: page 7
1.6 Simplifying Surds: page 9
1.7 Rationalising the Denominator: page 11
Exam revision: page 13
Memorandum: page 14
3
© 2026 Bluebird Notes. All rights reserved.
,Chapter 1
Exponents and Surds
Introduction
Mathematics is often about recognising patterns, and exponents are one of the clearest
examples of this. They provide a concise way of writing repeated multiplication, making
calculations simpler and allowing us to work with both extremely large and extremely small
numbers.
Surds extend this idea by introducing irrational numbers that cannot be written exactly as
fractions or terminating decimals. Although they may seem unfamiliar at first, surds allow us
to express exact values without rounding, making them essential in algebra, geometry and
many real world applications.
Throughout this chapter, you will learn how to simplify expressions involving exponents and
surds, apply the laws of exponents with confidence, rationalise denominators and solve
equations involving both concepts. By understanding the reasoning behind each rule, rather
than simply memorising formulas, you will develop the skills needed to approach
examination questions with confidence.
© Bluebird Notes
1.1 What is an Exponent?
Mathematics often uses shorthand notation to make calculations simpler. Instead of writing the
same number repeatedly, exponents provide a concise way to represent repeated multiplication.
For example, consider the expression:
5×5×5×5
Rather than writing the multiplication in full, it can be written more simply as:
54
This notation tells us that the number 5 is multiplied by itself four times.
Definition
An exponent, also known as an index or power, indicates how many times a number, called
the base, is multiplied by itself.
For the expression
54
4
© 2026 Bluebird Notes. All rights reserved.
, • 5 is the base.
• 4 is the exponent (or index).
This means:
54 = 5 × 5 × 5 × 5 = 625
Key Terms
Term Meaning
Base The number being multiplied repeatedly.
Exponent (Index) The number that indicates how many times
the base is used as a factor.
Power The complete exponential expression, for
example 54 .
Bluebird Tip© Bluebird Notes
Think of the exponent as an instruction, not a number to multiply by.
For example:
43
does not mean
4×3
Instead, it means
4 × 4 × 4.
The exponent tells you how many copies of the base to multiply together.
Common Mistake
Many students confuse
34
With
3 × 4.
Remember:
34 = 4 × 4 × 4 × 4 = 81
Whereas
5
© 2026 Bluebird Notes. All rights reserved.
, 3 × 4 = 12
These are completely different calculations.
Try it Yourself
Without using a calculator, write each expression as repeated multiplication and then
evaluate it.
1. 25
2. 72
3. 103
4. 61
1.2 Laws of Exponents
Law Rule Formula
Product law When multiplying powers 𝑎𝑚 × 𝑎𝑛 = 𝑎𝑚+𝑛
with the same base, add the
exponents.
Quotient Law When dividing powers with 𝑎𝑚
the same base, subtract the 𝑛
= 𝑎𝑚−𝑛
𝑎
exponents.
Power of a Power Multiply the exponents. (𝑎𝑚 )𝑛 = 𝑎𝑚𝑛
Power of a Product Apply the exponent to every (𝑎𝑏)𝑛 = 𝑎𝑛 𝑏𝑛
factor.
Power of a Quotient Apply the exponent to both 𝑎 𝑛 𝑎𝑛
the numerator and ( ) = 𝑛
𝑏 𝑏
denominator.
Zero Exponent Any non zero number raised 𝑎0 = 1 , 𝑎 ≠ 0
to the power of zero equals 1.
Negative Exponent Rewrite using the reciprocal 1
and make the exponent 𝑎−𝑛 = 𝑛 , 𝑎 ≠ 0
𝑎
positive.
🐦Remember © Bluebird Notes
• Only add or subtract exponents when the bases are the same.
• A negative exponent does not make the answer negative.
• The Zero Exponent Law only applies to non zero bases.
• Always simplify your final answer where possible.
1.3 Numerical Bases
Example 1 ⭐️
Simplify:
6
© 2026 Bluebird Notes. All rights reserved.
Grade 11 Mathematics
Chapter 1
Exponents and Surds
2026 Edition
Complete Chapter Notes for the IEB Curriculum
Comprehensive Theory • Worked Examples • Exam Tips
1
© 2026 Bluebird Notes. All rights reserved.
,Copyright
© 2026 Bluebird Notes
All rights reserved.
No part of this publication may be reproduced, distributed or shared in any form without prior
written permission from Bluebird Notes.
These notes are intended for personal educational use only.
Contact
Bluebird Notes
© Bluebird Notes
2
© 2026 Bluebird Notes. All rights reserved.
,Table of Contents
Chapter 1: page 4
1.1 What is an Exponent: page 4
1.2 Laws of Exponents: page 6
1.3 Numerical Bases: page 6
1.4 Expressions with Irrational Exponents: page 7
1.6 Simplifying Surds: page 9
1.7 Rationalising the Denominator: page 11
Exam revision: page 13
Memorandum: page 14
3
© 2026 Bluebird Notes. All rights reserved.
,Chapter 1
Exponents and Surds
Introduction
Mathematics is often about recognising patterns, and exponents are one of the clearest
examples of this. They provide a concise way of writing repeated multiplication, making
calculations simpler and allowing us to work with both extremely large and extremely small
numbers.
Surds extend this idea by introducing irrational numbers that cannot be written exactly as
fractions or terminating decimals. Although they may seem unfamiliar at first, surds allow us
to express exact values without rounding, making them essential in algebra, geometry and
many real world applications.
Throughout this chapter, you will learn how to simplify expressions involving exponents and
surds, apply the laws of exponents with confidence, rationalise denominators and solve
equations involving both concepts. By understanding the reasoning behind each rule, rather
than simply memorising formulas, you will develop the skills needed to approach
examination questions with confidence.
© Bluebird Notes
1.1 What is an Exponent?
Mathematics often uses shorthand notation to make calculations simpler. Instead of writing the
same number repeatedly, exponents provide a concise way to represent repeated multiplication.
For example, consider the expression:
5×5×5×5
Rather than writing the multiplication in full, it can be written more simply as:
54
This notation tells us that the number 5 is multiplied by itself four times.
Definition
An exponent, also known as an index or power, indicates how many times a number, called
the base, is multiplied by itself.
For the expression
54
4
© 2026 Bluebird Notes. All rights reserved.
, • 5 is the base.
• 4 is the exponent (or index).
This means:
54 = 5 × 5 × 5 × 5 = 625
Key Terms
Term Meaning
Base The number being multiplied repeatedly.
Exponent (Index) The number that indicates how many times
the base is used as a factor.
Power The complete exponential expression, for
example 54 .
Bluebird Tip© Bluebird Notes
Think of the exponent as an instruction, not a number to multiply by.
For example:
43
does not mean
4×3
Instead, it means
4 × 4 × 4.
The exponent tells you how many copies of the base to multiply together.
Common Mistake
Many students confuse
34
With
3 × 4.
Remember:
34 = 4 × 4 × 4 × 4 = 81
Whereas
5
© 2026 Bluebird Notes. All rights reserved.
, 3 × 4 = 12
These are completely different calculations.
Try it Yourself
Without using a calculator, write each expression as repeated multiplication and then
evaluate it.
1. 25
2. 72
3. 103
4. 61
1.2 Laws of Exponents
Law Rule Formula
Product law When multiplying powers 𝑎𝑚 × 𝑎𝑛 = 𝑎𝑚+𝑛
with the same base, add the
exponents.
Quotient Law When dividing powers with 𝑎𝑚
the same base, subtract the 𝑛
= 𝑎𝑚−𝑛
𝑎
exponents.
Power of a Power Multiply the exponents. (𝑎𝑚 )𝑛 = 𝑎𝑚𝑛
Power of a Product Apply the exponent to every (𝑎𝑏)𝑛 = 𝑎𝑛 𝑏𝑛
factor.
Power of a Quotient Apply the exponent to both 𝑎 𝑛 𝑎𝑛
the numerator and ( ) = 𝑛
𝑏 𝑏
denominator.
Zero Exponent Any non zero number raised 𝑎0 = 1 , 𝑎 ≠ 0
to the power of zero equals 1.
Negative Exponent Rewrite using the reciprocal 1
and make the exponent 𝑎−𝑛 = 𝑛 , 𝑎 ≠ 0
𝑎
positive.
🐦Remember © Bluebird Notes
• Only add or subtract exponents when the bases are the same.
• A negative exponent does not make the answer negative.
• The Zero Exponent Law only applies to non zero bases.
• Always simplify your final answer where possible.
1.3 Numerical Bases
Example 1 ⭐️
Simplify:
6
© 2026 Bluebird Notes. All rights reserved.
