Powers and Roots: The Foundation of Fast Calculation (0607 Number Section)
Hello mathematicians! This chapter, "Powers and Roots," is a fundamental building block in the "Number" section of your syllabus. Why is it so important? Because powers (or indices) are the mathematical shorthand for repeated multiplication, and roots are the way we undo them. Mastering this saves time, prevents calculator errors, and prepares you for complex algebra later on!
Don't worry if indices seem confusing—we'll break down the rules one by one, making sure even the trickiest concepts like negative and fractional powers become simple. Let’s power up your knowledge!
1. Core Concepts: Squares, Cubes, and Roots (C1.3)
1.1 Powers: The Shorthand for Repeated Multiplication
When we talk about a power, we are multiplying a base number by itself a specific number of times, indicated by the small raised number, called the index (or exponent).
For example, in \(5^3\):
The base is 5.
The index (or power) is 3.
\(5^3 = 5 \times 5 \times 5 = 125\).
Key Power Terms:
- Square numbers: A number raised to the power of 2 (index is 2). This gives you the area of a square. Example: \(4^2 = 16\).
- Cube numbers: A number raised to the power of 3 (index is 3). This gives you the volume of a cube. Example: \(4^3 = 64\).
- Other powers: \(4^5\) (four to the power of five) is \(4 \times 4 \times 4 \times 4 \times 4\).
1.2 Roots: Undoing the Power
A root is the opposite operation of a power. It asks: "What number, when multiplied by itself (a certain number of times), gives this result?"
- Square Root (\(\sqrt{}\)): The opposite of squaring. Example: \(\sqrt{16} = 4\) because \(4 \times 4 = 16\).
- Cube Root (\(\sqrt[3]{}\)): The opposite of cubing. Example: \(\sqrt[3]{64} = 4\) because \(4 \times 4 \times 4 = 64\).
- Other Roots (\(\sqrt[n]{}\)): For example, the fourth root (\(\sqrt[4]{16}\)) is 2, because \(2^4 = 16\).
Memory Aid (C1.3 Required Recall): You should recall (know quickly) the squares up to \(15^2\) and the corresponding square roots, and the cubes and cube roots of 1, 2, 3, 4, 5, and 10.
Example task: Work out \(5^2 \times \sqrt[3]{8}\).
Step 1: Calculate the power: \(5^2 = 25\).
Step 2: Calculate the root: \(\sqrt[3]{8} = 2\) (since \(2 \times 2 \times 2 = 8\)).
Step 3: Multiply: \(25 \times 2 = 50\).
Key Takeaway (Section 1): Powers and roots are inverse operations. Know your common squares and cubes instantly!
2. Indices: Extending Powers to Zero, Negatives, and Fractions (C1.7 / E1.7)
The rules of indices allow us to use positive, negative, and even fractional numbers as powers. This is where maths gets really powerful!
2.1 The Zero Index Rule
Rule: Anything (except 0) raised to the power of zero equals 1.
\[a^0 = 1\]
Example: \(100^0 = 1\), \((-5)^0 = 1\), \((x^2)^0 = 1\).
Analogy: Imagine your index is a marker showing how many items you have. If you have \(a^3\), you have three copies of 'a'. If you have \(a^0\), you haven't multiplied 'a' at all, so you just have the 'start' value, which is 1 (like having a placeholder).
2.2 The Negative Index Rule
Rule: A negative index means "take the reciprocal" of the base raised to the corresponding positive power.
\[a^{-n} = \frac{1}{a^n}\]
Example: \(5^{-2} = \frac{1}{5^2} = \frac{1}{25}\).
Common Mistake to Avoid: A negative index does NOT make the answer negative. It flips the number!
Example task: Find the value of \(7^{-2}\).
Solution: \(7^{-2} = \frac{1}{7^2} = \frac{1}{49}\).
2.3 Fractional Indices (Extended Content E1.7)
This rule connects indices directly to roots. The denominator (bottom) of the fraction tells you the root, and the numerator (top) tells you the power.
\[a^{\frac{1}{n}} = \sqrt[n]{a}\]
\[a^{\frac{m}{n}} = (\sqrt[n]{a})^m \text{ or } \sqrt[n]{(a^m)}\]
- If the power is \(\frac{1}{2}\), it means square root: \(16^{\frac{1}{2}} = \sqrt{16} = 4\).
- If the power is \(\frac{1}{3}\), it means cube root: \(8^{\frac{1}{3}} = \sqrt[3]{8} = 2\).
When the numerator is not 1, deal with the root first, as the numbers usually stay smaller and easier to manage.
Example: Work out \(27^{\frac{2}{3}}\).
Step 1 (Root): Find the cube root: \(\sqrt[3]{27} = 3\).
Step 2 (Power): Square the result: \(3^2 = 9\).
Solution: \(27^{\frac{2}{3}} = 9\).
Key Takeaway (Section 2): \(a^0 = 1\). Negative powers mean reciprocals. Fractional powers mean roots (denominator) and powers (numerator).
3. The Rules for Calculating with Indices (C1.7 / E1.7 / E2.4)
These rules let you simplify expressions quickly without calculating the actual values, which is essential for algebra.
3.1 Multiplication Rule (Adding Powers)
Rule: When multiplying terms with the same base, add the indices.
\[a^m \times a^n = a^{m+n}\]
Think: \(x^2 \times x^3 = (x \times x) \times (x \times x \times x) = x^5\) (2 + 3 = 5).
Example: \(2^{-3} \times 2^4 = 2^{(-3+4)} = 2^1 = 2\).
3.2 Division Rule (Subtracting Powers)
Rule: When dividing terms with the same base, subtract the indices.
\[a^m \div a^n = a^{m-n}\]
Think: \(\frac{x^5}{x^2} = \frac{x \times x \times x \times x \times x}{x \times x} = x^3\) (5 - 2 = 3).
Example: \(2^3 \div 2^4 = 2^{(3-4)} = 2^{-1} = \frac{1}{2}\).
3.3 Power of a Power Rule (Multiplying Powers)
Rule: When raising a power to another power, multiply the indices.
\[(a^m)^n = a^{m \times n}\]
Think: \((x^2)^3 = x^2 \times x^2 \times x^2 = x^{2+2+2} = x^6\).
Example: \((2^3)^2 = 2^6 = 64\).
3.4 Power of a Product Rule
Rule: When raising a product to a power, raise each factor to that power.
\[(ab)^n = a^n b^n\]
Example: Simplify \((5x^3)^2\).
Solution: \((5x^3)^2 = 5^2 \times (x^3)^2 = 25 x^6\).
Quick Review Box for Index Rules
- Multiply: Add powers
- Divide: Subtract powers
- Power of Power: Multiply powers
- Negative Power: Reciprocal (Flip it!)
- Zero Power: Equals 1
Key Takeaway (Section 3): These three core operations (multiplication, division, power of a power) form the basis of all index simplifications. Be careful to apply the power to all parts of a product or fraction.
4. Working with Surds (Extended Content E1.17)
Sometimes when you take a root, the answer isn't a neat whole number or a fraction—it’s an irrational number. When we leave an irrational root in its exact form (like \(\sqrt{2}\) instead of 1.414...), we call it a surd. Surds are part of the extended content and allow us to give mathematically accurate "exact" answers.
4.1 What is a Surd?
A surd is an expression involving the root of a number (usually a square root) that cannot be simplified to a whole number or a rational fraction.
- \(\sqrt{9} = 3\). Not a surd (it's rational).
- \(\sqrt{7}\). This is a surd (it's irrational).
4.2 Simplifying Surds
We simplify surds by finding the largest possible square number factor within the root.
Step-by-step Example: Simplify \(\sqrt{20}\)
- Find the largest square number that divides into 20. The square numbers are 4, 9, 16, 25... The largest one is 4.
- Rewrite the number as a product: \(\sqrt{20} = \sqrt{4 \times 5}\).
- Split the root: \(\sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5}\).
- Simplify the square root: \(2 \times \sqrt{5}\).
Solution: \(\sqrt{20} = 2\sqrt{5}\).
4.3 Addition and Subtraction of Surds
You can only add or subtract like surds (those with the same number inside the root). It's just like collecting like terms in algebra!
Example: Simplify \(\sqrt{200} - \sqrt{32}\)
- Simplify \(\sqrt{200}\): \(\sqrt{100 \times 2} = 10\sqrt{2}\).
- Simplify \(\sqrt{32}\): \(\sqrt{16 \times 2} = 4\sqrt{2}\).
- Subtract the like surds: \(10\sqrt{2} - 4\sqrt{2} = 6\sqrt{2}\).
4.4 Rationalising the Denominator
Mathematicians prefer fractions not to have surds in the denominator. The process of removing the surd from the bottom of the fraction is called rationalising the denominator.
Case 1: Single Surd in the Denominator
Multiply the top and bottom of the fraction by the surd itself. This uses the rule that \(\sqrt{a} \times \sqrt{a} = a\).
Example: Rationalise \(\frac{10}{\sqrt{5}}\).
\[\frac{10}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{10\sqrt{5}}{5}\]
Simplify the fraction: \(\frac{10}{5} = 2\).
Solution: \(2\sqrt{5}\).
Case 2: Binomial Surd in the Denominator (Extended Challenge)
If the denominator is in the form \(a + \sqrt{b}\) or \(a - \sqrt{b}\), we multiply by its conjugate. The conjugate has the same terms but the opposite sign in the middle (e.g., the conjugate of \(a+\sqrt{b}\) is \(a-\sqrt{b}\)).
Example (from E1.17 Notes): Rationalise \(\frac{1}{-1+\sqrt{3}}\).
The conjugate of \(-1 + \sqrt{3}\) is \(-1 - \sqrt{3}\) or, more neatly, \(\sqrt{3} - 1\). Let's use \(\sqrt{3} - 1\). The conjugate 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 (using Difference of Two Squares: \((a-b)(a+b) = a^2 - b^2\)):
\((\sqrt{3})^2 - (1)^2 = 3 - 1 = 2\)
Solution: \(\frac{1 + \sqrt{3}}{2}\). (Note: The syllabus example is given as \(\frac{1+\sqrt{3}}{2}\), which is correct.)
Did you know? Surds, like \(\sqrt{2}\) and \(\pi\), cannot be written as a terminating or repeating decimal. This concept caused serious mathematical drama thousands of years ago when the Greeks discovered numbers that couldn't be expressed as ratios (fractions)!
Key Takeaway (Section 4): Surds are exact, irrational roots. Simplify by factoring square numbers. Rationalise the denominator using multiplication by the surd or the conjugate.