🚀 Chapter 1: Number – Powers and Roots (Indices)
Hello Mathematicians! Welcome to one of the most powerful (pun intended!) chapters in Number theory: Powers and Roots. This topic is all about mathematical shortcuts—how to write very big or very small numbers compactly, and how to handle repetitive multiplication easily.
Mastering powers and roots (also called indices or exponents) will unlock faster calculations and form the foundation for many topics later in Algebra and beyond. Don't worry if the rules seem tricky at first; with practice, they become second nature!
Key Terminology: Base and Index
When we write \(a^n\):
- Base (a): The number being multiplied.
- Index or Exponent (n): The number of times the base is multiplied by itself.
Example: In \(5^3\), the base is 5 and the index is 3. This means \(5 \times 5 \times 5 = 125\).
1. Basic Powers and Roots (Core Content C1.3)
We start with the most common types of powers you should recognise instantly.
1.1 Squares and Square Roots
A square number is the result of multiplying a number by itself.
\(a^2 = a \times a\)
The square root (\(\sqrt{\text{ }}\)) is the inverse operation—finding the number that was squared to get the result.
🧠 Memory Aid: Essential Squares and Roots
The syllabus requires you to recall squares and their roots from 1 to 15. Make sure you know these by heart!
- \(8^2 = 64\), so \(\sqrt{64} = 8\).
- \(12^2 = 144\), so \(\sqrt{144} = 12\).
- Example: Write down the value of \(\sqrt{169}\). (Answer: 13)
Did you know? A perfect square always represents the area of a square shape where the side length is the square root.
1.2 Cubes and Cube Roots
A cube number is the result of multiplying a number by itself three times.
\(a^3 = a \times a \times a\)
The cube root (\(\sqrt[3]{\text{ }}\)) is the inverse operation.
🧠 Memory Aid: Essential Cubes and Roots
The syllabus requires you to recall cubes and their roots for 1, 2, 3, 4, 5, and 10.
- \(2^3 = 8\), so \(\sqrt[3]{8} = 2\).
- \(5^3 = 125\), so \(\sqrt[3]{125} = 5\).
- Example: Work out \(5^2 \times \sqrt[3]{8}\).
Step 1: \(5^2 = 25\).
Step 2: \(\sqrt[3]{8} = 2\).
Step 3: \(25 \times 2 = 50\).
1.3 General Powers and Roots
You will also work with higher powers, such as fourth power (\(a^4\)) or fifth root (\(\sqrt[5]{\text{ }}\)). The concept remains the same:
- Power: \(a^n\) means multiplying a by itself n times.
- Root: \(\sqrt[n]{a}\) means finding the number that, when multiplied by itself n times, gives a.
Key Takeaway (Section 1)
Squares and cubes are fundamental powers, and their roots are the reverse operation. Practice recalling the key values up to \(15^2\) and \(10^3\).
2. The Rules of Indices (Core Content C1.7 & Extended Content E1.7)
The true power of indices lies in the shortcuts (rules) we can use when multiplying, dividing, or raising powers to other powers. These rules apply to positive, zero, and negative integer indices in Core Maths, and fractional indices in Extended Maths.
2.1 Rule 1: Multiplication (\(a^m \times a^n\))
When multiplying powers with the same base, you add the indices.
$$a^m \times a^n = a^{m+n}$$
- Simple Example: \(2^3 \times 2^4 = 2^{3+4} = 2^7\). (Check: \(8 \times 16 = 128\), and \(2^7 = 128\).)
- Algebraic Example: \(x^5 \times x^2 = x^7\).
2.2 Rule 2: Division (\(a^m \div a^n\))
When dividing powers with the same base, you subtract the indices.
$$\frac{a^m}{a^n} = a^{m-n}$$
- Simple Example: \(3^5 \div 3^2 = 3^{5-2} = 3^3 = 27\).
- Syllabus Example: \(2^3 \div 2^4 = 2^{3-4} = 2^{-1}\). (This leads nicely to the negative index rule!)
2.3 Rule 3: Power of a Power (\((a^m)^n\))
When raising a power to another power, you multiply the indices.
$$(a^m)^n = a^{m \times n}$$
- Simple Example: \((5^2)^3 = 5^{2 \times 3} = 5^6\).
- Syllabus Example: Simplify \((2^3)^2\). (Answer: \(2^6\)).
🛑 Common Mistake Alert!
Do not confuse \((a^m)^n\) with \(a^m \times a^n\)!
\((x^2)^3 = x^6\) (multiply powers)
\(x^2 \times x^3 = x^5\) (add powers)
2.4 Rule 4: The Zero Index (\(a^0\))
Any number (except zero) raised to the power of zero is always 1.
$$a^0 = 1 \quad (\text{where } a \ne 0)$$
- Example: \(100^0 = 1\).
- Example: \((5x)^0 = 1\).
2.5 Rule 5: The Negative Index (\(a^{-n}\))
A negative index means "take the reciprocal" of the base raised to the positive power. It tells you to move the expression across the fraction line.
$$a^{-n} = \frac{1}{a^n}$$
- Simple Example: \(7^{-2} = \frac{1}{7^2} = \frac{1}{49}\). (Syllabus example: Find the value of \(7^{-2}\)).
- Fractional Example: \(\left(\frac{2}{3}\right)^{-1} = \frac{3}{2}\). (Just flip the fraction!)
Quick Review: Rules of Indices
Multiply (Same Base): Add powers (M.A.)
Divide (Same Base): Subtract powers (D.S.)
Power of a Power: Multiply powers (P.M.)
Zero Power: Equals 1
Negative Power: Flip and Power
3. Fractional Indices (Extended Content E1.7)
This is where indices meet roots! If the index is a fraction, it represents a root.
3.1 The Rule for Fractional Indices
The denominator of the fraction is the root, and the numerator is the power.
$$a^{\frac{m}{n}} = \sqrt[n]{a^m} \quad \text{or} \quad (\sqrt[n]{a})^m$$
Don't worry, the formula looks scarier than it is! Usually, you calculate the root first, as it makes the number smaller and easier to work with.
- Simple Root Example: \(16^{\frac{1}{2}} = \sqrt[2]{16} = 4\). (The index \(\frac{1}{2}\) means square root).
- Cube Root Example: \(8^{\frac{1}{3}} = \sqrt[3]{8} = 2\).
3.2 Using Fractional Indices for Calculation
When the numerator is not 1, break the calculation into two parts: root first, then power.
- Example: Calculate \(27^{\frac{2}{3}}\).
Step 1 (Root): Find the cube root (denominator = 3): \(\sqrt[3]{27} = 3\).
Step 2 (Power): Raise the result to the power of 2 (numerator = 2): \(3^2 = 9\).
So, \(27^{\frac{2}{3}} = 9\).
We can combine this with the negative index rule:
- Example: Calculate \(8^{-\frac{2}{3}}\).
Step 1 (Negative): Flip the expression: \(\frac{1}{8^{\frac{2}{3}}}\).
Step 2 (Root): Calculate \(8^{\frac{2}{3}}\). Cube root of 8 is 2. \(2^2 = 4\).
Final Answer: \(\frac{1}{4}\).
Key Takeaway (Section 3)
Fractional indices are just a different way of writing roots. \(a^{\frac{1}{n}}\) means the \(n\)-th root of \(a\). If the index is negative, flip the fraction first!
4. Introduction to Surds (Extended Content E1.17)
Sometimes, when you take a root, the answer isn't a neat whole number or a clean fraction (it’s irrational). These messy, inexact roots are called surds.
A surd is an irrational number that is the result of taking the root of an integer.
- Example of a Surd: \(\sqrt{2}\) (\(\approx 1.4142...\))
- Example that is NOT a Surd: \(\sqrt{9} = 3\) (This is a rational number).
4.1 Simplifying Surds
We simplify surds by finding the largest perfect square factor inside the root sign.
Rule: \(\sqrt{ab} = \sqrt{a} \times \sqrt{b}\)
Step-by-Step Simplification
Example: Simplify \(\sqrt{20}\).
- Find the largest perfect square that divides 20. The factors are 1, 2, 4, 5, 10, 20. The largest square is 4.
- Rewrite the surd using this factor: \(\sqrt{20} = \sqrt{4 \times 5}\).
- Separate and simplify the square root: \(\sqrt{4} \times \sqrt{5} = 2\sqrt{5}\).
Syllabus Example: Simplify \(\sqrt{200} - \sqrt{32}\).
- \(\sqrt{200} = \sqrt{100 \times 2} = 10\sqrt{2}\)
- \(\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}\)
- Subtraction (only possible because the surds are the same, like terms in algebra): \(10\sqrt{2} - 4\sqrt{2} = 6\sqrt{2}\).
4.2 Rationalising the Denominator
In Mathematics, we often prefer the denominator of a fraction not to contain a surd (an irrational number). The process of removing the surd from the denominator is called rationalising.
Case 1: Single Surd in the Denominator
Multiply the numerator and denominator by the surd itself.
Example: Rationalise \(\frac{10}{\sqrt{5}}\).
$$\frac{10}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{10\sqrt{5}}{5}$$
Now, simplify the numbers: \(\frac{10}{5} = 2\).
Final Answer: \(2\sqrt{5}\). (This matches the syllabus example: \(\frac{10}{\sqrt{5}} = 2\sqrt{5}\)).
Case 2: Binomial Surd in the Denominator (Using the Conjugate)
If the denominator looks like \(a + \sqrt{b}\) or \(a - \sqrt{b}\), you must multiply by the conjugate. The conjugate is the expression with the middle sign flipped.
This works because \((x+y)(x-y) = x^2 - y^2\), which eliminates the square root!
Syllabus Example: Rationalise \(\frac{1}{-1+\sqrt{3}}\). (The terms can be rearranged to \(\sqrt{3} - 1\)).
The conjugate of \(\sqrt{3} - 1\) is \(\sqrt{3} + 1\).
$$\frac{1}{\sqrt{3} - 1} \times \frac{\sqrt{3} + 1}{\sqrt{3} + 1}$$
Numerator: \(1 \times (\sqrt{3} + 1) = \sqrt{3} + 1\)
Denominator: \((\sqrt{3} - 1)(\sqrt{3} + 1) = (\sqrt{3})^2 - (1)^2 = 3 - 1 = 2\)
Final Answer: \(\frac{\sqrt{3} + 1}{2}\)
Key Takeaway (Section 4)
Surds are irrational roots. Simplify them by extracting square factors. Rationalising removes surds from the denominator; use the conjugate when the denominator is a binomial (two terms involving a surd).