Integers, powers and roots

Q: What are the rules for multiplying or dividing directed numbers (integers) with the same or different signs?

Same signs result in a positive answer, while different signs result in a negative answer.

Q: What does the BIDMAS acronym stand for and what is it used for?

BIDMAS defines the correct order of operations: B rackets, I ndices, D ivision & M ultiplication, A ddition & S ubtraction.

Q: What is a prime number, and why are 1 and 2 unique?

A prime number is a whole number greater than \( 1 \) with exactly two distinct factors: \( 1 \) and itself. \( 1 \) is not prime, and \( 2 \) is the only even prime number.

Q: What is prime factor decomposition?

It is the process of writing any integer greater than \( 1 \) as a unique product of its prime factors.

Q: How do you find the HCF and LCM of two numbers using a Venn diagram of their prime factors?

The HCF is the product of the shared prime factors in the intersection, and the LCM is the product of all prime factors across the entire Venn diagram.

Q: What is the rule for multiplying and dividing terms with the same base?

When multiplying, add the indices: \( a^m \times a^n = a^{m+n} \). When dividing, subtract the indices: \( \frac{a^m}{a^n} = a^{m-n} \).

Q: How do you evaluate an expression with a power of zero or a negative index?

Any non-zero base to the power of \( 0 \) equals \( 1 \) (\( a^0 = 1 \)), and a negative index indicates the reciprocal (\( a^{-n} = \frac{1}{a^n} \)).

Q: How do fractional indices relate to roots and powers?

The denominator of the fraction represents the root, and the numerator represents the power: \( a^{\frac{m}{n}} = (\sqrt[n]{a})^m \).

Q: What is the difference between finding the square root and the cube root of a negative number?

Square roots of negative numbers are not real numbers, but cube roots of negative numbers are negative (e.g., \( \sqrt[3]{-8} = -2 \)).

Q: How do you estimate the square root of a non-square number without a calculator?

Find the two consecutive square numbers it falls between, take their square roots, and estimate the decimal based on proximity.

Mathematics 2210 · CCEA GCSE · 10 min read · Updated: 31 Aug 2026

Welcome to Integers, Powers, and Roots!

Welcome to one of the most important foundations in GCSE Mathematics! In this chapter, we explore how numbers behave, from everyday negative numbers like temperatures below freezing, to repeated multiplication using powers, and reversing the process with roots. Don't worry if you have found these tricky in the past; we will break down every single idea step by step so you can master them with confidence.

1. Understanding Integers and Directed Numbers

An integer is simply any whole number. Integers can be positive, negative, or zero: \( \dots, -3, -2, -1, 0, 1, 2, 3, \dots \). Decimals and fractions like \( 0.5 \) or \( \frac{3}{4} \) are not integers.

Working with Negative Numbers

Think of directed numbers using a temperature or bank balance analogy: positive is money you have or degrees above zero, while negative is money you owe or degrees below zero.

Adding and Subtracting:
• Adding a positive number moves you right (up) the number line: \( -3 + 5 = 2 \)
• Subtracting a positive number moves you left (down) the number line: \( 2 - 6 = -4 \)
• When two signs appear next to each other:
    • \( + \) and \( + \) become \( + \)   \( \implies 4 + (+3) = 4 + 3 = 7 \)
    • \( + \) and \( - \) become \( - \)   \( \implies 5 + (-2) = 5 - 2 = 3 \)
    • \( - \) and \( + \) become \( - \)   \( \implies 5 - (+2) = 5 - 2 = 3 \)
    • \( - \) and \( - \) become \( + \)   \( \implies 4 - (-3) = 4 + 3 = 7 \)

Memory Trick: If the two signs next to each other are the same, replace them with a plus. If they are different, replace them with a minus.

Multiplying and Dividing:
• Same signs give a positive answer:
    \( 4 \times 3 = 12 \)
    \( (-4) \times (-3) = 12 \)
    \( (-20) \div (-4) = 5 \)
• Different signs give a negative answer:
    \( (-4) \times 3 = -12 \)
    \( 4 \times (-3) = -12 \)
    \( 20 \div (-4) = -5 \)

Order of Operations (BIDMAS)

When solving calculations with multiple operations, always follow the BIDMAS rule:
• Brackets: \( (2 + 3) \)
• Indices (Powers / Roots): \( 4^2 \) or \( \sqrt{9} \)
• Division and Multiplication (work from left to right)
• Addition and Subtraction (work from left to right)

Example: Calculate \( 3 + 2 \times 4^2 \)
Step 1 (Indices): \( 4^2 = 16 \)
Step 2 (Multiplication): \( 2 \times 16 = 32 \)
Step 3 (Addition): \( 3 + 32 = 35 \)

Section 1 Key Takeaway

Integers are whole numbers. When multiplying or dividing, two like signs make a positive, and two different signs make a negative. Always follow BIDMAS.

2. Factors, Multiples, and Prime Numbers

Definitions

• Factor: A whole number that divides exactly into another number without leaving a remainder. For example, the factors of \( 12 \) are \( 1, 2, 3, 4, 6, 12 \).
• Multiple: The result of multiplying a number by an integer (think of your times tables). For example, the multiples of \( 5 \) are \( 5, 10, 15, 20, 25, \dots \)
• Prime Number: A whole number greater than \( 1 \) that has exactly two factors: \( 1 \) and itself. The first few prime numbers are \( 2, 3, 5, 7, 11, 13, 17, 19, 23 \).

Did you know? The number \( 1 \) is not a prime number because it only has one factor (itself), and \( 2 \) is the only even prime number!

Prime Factor Decomposition

Every integer greater than \( 1 \) can be broken down into a unique product of prime numbers. We often use a factor tree to do this.

Step-by-Step Example: Express \( 60 \) as a product of prime factors
1. Pick any two numbers that multiply to give \( 60 \), for example \( 6 \times 10 \).
2. Split \( 6 \) into \( 2 \times 3 \). Both are prime, so circle them.
3. Split \( 10 \) into \( 2 \times 5 \). Both are prime, so circle them.
4. Write out the circled numbers: \( 60 = 2 \times 2 \times 3 \times 5 \)
5. In index form: \( 60 = 2^2 \times 3 \times 5 \)

Finding HCF and LCM Using Prime Factors

• Highest Common Factor (HCF): The largest number that divides evenly into two or more numbers.
• Lowest Common Multiple (LCM): The smallest number that is a multiple of two or more numbers.

Example: Find the HCF and LCM of \( 24 \) and \( 60 \)
1. Prime factors of \( 24 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3 \)
2. Prime factors of \( 60 = 2 \times 2 \times 3 \times 5 = 2^2 \times 3 \times 5 \)
3. Place them into a Venn Diagram:
    • Shared factors (Intersection): \( 2, 2, 3 \)
    • Remaining factors for \( 24 \): \( 2 \)
    • Remaining factors for \( 60 \): \( 5 \)
4. HCF = Multiply the shared factors in the overlap: \( 2 \times 2 \times 3 = 12 \)
5. LCM = Multiply ALL numbers in the Venn diagram: \( 2 \times (2 \times 2 \times 3) \times 5 = 120 \)

Section 2 Key Takeaway

Prime factors are the basic building blocks of numbers. Use factor trees to find prime factorisations, then multiply the common primes for HCF, and all primes across the diagram for LCM.

3. Powers (Indices)

A power (or index / exponent) shows how many times a number is multiplied by itself.

In the expression \( 2^4 \):
• \( 2 \) is the base
• \( 4 \) is the index (or power)
• \( 2^4 = 2 \times 2 \times 2 \times 2 = 16 \)

The Laws of Indices

These rules make working with powers fast and simple. Learn them well!

Rule 1: Multiplication Law (Add the powers)
When multiplying terms with the same base, add the indices:
\( a^m \times a^n = a^{m+n} \)
Example: \( 3^2 \times 3^4 = 3^{2+4} = 3^6 \)

Rule 2: Division Law (Subtract the powers)
When dividing terms with the same base, subtract the bottom index from the top index:
\( \frac{a^m}{a^n} = a^{m-n} \)
Example: \( \frac{5^7}{5^3} = 5^{7-3} = 5^4 \)

Rule 3: Power of a Power Law (Multiply the powers)
When raising a power to another power, multiply the indices:
\( (a^m)^n = a^{m \times n} \)
Example: \( (2^3)^4 = 2^{3 \times 4} = 2^{12} \)

Rule 4: Power of Zero
Any non-zero number raised to the power of \( 0 \) is equal to \( 1 \):
\( a^0 = 1 \)
Example: \( 7^0 = 1 \), \( 100^0 = 1 \), \( x^0 = 1 \)

Rule 5: Negative Indices (The Reciprocal Rule)
A negative power tells you to turn the number into its reciprocal (put it under \( 1 \)):
\( a^{-n} = \frac{1}{a^n} \)
Example: \( 5^{-2} = \frac{1}{5^2} = \frac{1}{25} \)
Example: \( 2^{-3} = \frac{1}{2^3} = \frac{1}{8} \)

Rule 6: Fractional Indices (Roots)
The denominator of a fraction indicates the root, and the numerator indicates the power:
\( a^{\frac{1}{n}} = \sqrt[n]{a} \)
\( a^{\frac{m}{n}} = (\sqrt[n]{a})^m \)
Example: \( 16^{\frac{1}{2}} = \sqrt{16} = 4 \)
Example: \( 27^{\frac{1}{3}} = \sqrt[3]{27} = 3 \)
Example: \( 8^{\frac{2}{3}} = (\sqrt[3]{8})^2 = 2^2 = 4 \)

Section 3 Key Takeaway

Keep the bases the same to apply index rules: multiply means add indices, divide means subtract indices, brackets mean multiply indices, negative powers mean take the reciprocal, and fractional powers mean find the root.

4. Roots (Square Roots and Cube Roots)

A root is the inverse (opposite) operation of a power.

Square Roots

The square root of a number \( x \), written as \( \sqrt{x} \), is the value that multiplies by itself to give \( x \).
• \( 6^2 = 36 \implies \sqrt{36} = 6 \)
• Square Numbers to Memorise:
\( 1^2 = 1 \), \( 2^2 = 4 \), \( 3^2 = 9 \), \( 4^2 = 16 \), \( 5^2 = 25 \), \( 6^2 = 36 \), \( 7^2 = 49 \), \( 8^2 = 64 \), \( 9^2 = 81 \), \( 10^2 = 100 \), \( 11^2 = 121 \), \( 12^2 = 144 \), \( 13^2 = 169 \), \( 14^2 = 196 \), \( 15^2 = 225 \).

Important Note: While \( (-6) \times (-6) = 36 \), the radical sign \( \sqrt{36} \) refers specifically to the principal (positive) square root, which is \( 6 \). However, when solving equations like \( x^2 = 36 \), there are two solutions: \( x = \pm 6 \).

Cube Roots

The cube root of a number \( x \), written as \( \sqrt[3]{x} \), is the number that must be multiplied by itself three times to get \( x \).
• \( 2^3 = 2 \times 2 \times 2 = 8 \implies \sqrt[3]{8} = 2 \)
• Cube Numbers to Memorise:
\( 1^3 = 1 \), \( 2^3 = 8 \), \( 3^3 = 27 \), \( 4^3 = 64 \), \( 5^3 = 125 \), \( 10^3 = 1000 \).
• Unlike square roots, you can take the cube root of a negative number: \( \sqrt[3]{-8} = -2 \) because \( (-2) \times (-2) \times (-2) = -8 \).

Estimating Roots

When dealing with non-square numbers, find the two square numbers it sits between.
Example: Estimate the value of \( \sqrt{40} \) to one decimal place.
1. Identify surrounding square numbers: \( 36 < 40 < 49 \)
2. Take square roots: \( \sqrt{36} < \sqrt{40} < \sqrt{49} \implies 6 < \sqrt{40} < 7 \)
3. \( 40 \) is slightly closer to \( 36 \) than \( 49 \), so \( \sqrt{40} \approx 6.3 \).

Section 4 Key Takeaway

Roots undo powers. Square roots undo squaring (\( \sqrt{a} = a^{\frac{1}{2}} \)) and cube roots undo cubing (\( \sqrt[3]{a} = a^{\frac{1}{3}} \)).

5. Common Mistakes to Avoid

• Mistake 1: Writing \( 3^3 = 9 \).
Correction: \( 3^3 \) means \( 3 \times 3 \times 3 = 27 \), not \( 3 \times 3 \).
• Mistake 2: Thinking \( 2^{-3} = -6 \) or \( -8 \).
Correction: Negative powers do not make the number negative! \( 2^{-3} = \frac{1}{2^3} = \frac{1}{8} \).
• Mistake 3: Confusing \( (-3)^2 \) and \( -3^2 \).
Correction: \( (-3)^2 = (-3) \times (-3) = 9 \), but \( -3^2 = -(3 \times 3) = -9 \). Brackets matter!
• Mistake 4: Adding bases during index multiplication, e.g., \( 2^3 \times 2^4 = 4^7 \).
Correction: The base stays the same: \( 2^3 \times 2^4 = 2^{3+4} = 2^7 \).

6. Quick Review Checklist

Before your exam, make sure you can:
• Add, subtract, multiply, and divide positive and negative integers.
• Apply BIDMAS in multi-step calculations.
• Break any number down into its prime factors using a factor tree.
• Calculate the HCF and LCM of two numbers using prime factorisation.
• Use the index laws for multiplying, dividing, and raising powers to powers.
• Evaluate expressions with zero, negative, and fractional indices.
• Recall common square numbers up to \( 15^2 \) and cube numbers up to \( 5^3 \).

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What are the rules for multiplying or dividing directed numbers (integers) with the same or different signs?

Same signs result in a positive answer, while different signs result in a negative answer.

What does the BIDMAS acronym stand for and what is it used for?

BIDMAS defines the correct order of operations: Brackets, Indices, Division & Multiplication, Addition & Subtraction.

What is a prime number, and why are 1 and 2 unique?

A prime number is a whole number greater than \( 1 \) with exactly two distinct factors: \( 1 \) and itself. \( 1 \) is not prime, and \( 2 \) is the only even prime number.

What is prime factor decomposition?

It is the process of writing any integer greater than \( 1 \) as a unique product of its prime factors.

How do you find the HCF and LCM of two numbers using a Venn diagram of their prime factors?

The HCF is the product of the shared prime factors in the intersection, and the LCM is the product of all prime factors across the entire Venn diagram.

What is the rule for multiplying and dividing terms with the same base?

When multiplying, add the indices: \( a^m \times a^n = a^{m+n} \). When dividing, subtract the indices: \( \frac{a^m}{a^n} = a^{m-n} \).

How do you evaluate an expression with a power of zero or a negative index?

Any non-zero base to the power of \( 0 \) equals \( 1 \) (\( a^0 = 1 \)), and a negative index indicates the reciprocal (\( a^{-n} = \frac{1}{a^n} \)).

How do fractional indices relate to roots and powers?

The denominator of the fraction represents the root, and the numerator represents the power: \( a^{\frac{m}{n}} = (\sqrt[n]{a})^m \).

What is the difference between finding the square root and the cube root of a negative number?

Square roots of negative numbers are not real numbers, but cube roots of negative numbers are negative (e.g., \( \sqrt[3]{-8} = -2 \)).

How do you estimate the square root of a non-square number without a calculator?

Find the two consecutive square numbers it falls between, take their square roots, and estimate the decimal based on proximity.

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