1. Overview
Indices (also known as powers or exponents) are a shorthand way of expressing repeated multiplication. This revision note focuses on extending your understanding of indices to include zero, negative, and fractional powers. Mastering these concepts is crucial for simplifying algebraic expressions, solving equations, and tackling more advanced mathematical problems in the IGCSE Cambridge Mathematics (0580) syllabus.
Key Definitions
- Base: The number or variable being multiplied by itself (e.g., in $x^5$, $x$ is the base).
- Index (Exponent/Power): The number indicating how many times to multiply the base by itself (e.g., in $x^5$, $5$ is the index).
- Reciprocal: The value obtained by dividing 1 by a number; particularly relevant for negative indices (e.g., the reciprocal of $a$ is $\frac{1}{a}$).
- Root: The inverse operation of a power, denoted by the radical symbol $\sqrt[n]{}$. For example, $\sqrt[3]{8} = 2$ because $2^3 = 8$.
Core Content
There are no Core-specific objectives for this sub-topic; all learning objectives are part of the Supplement curriculum.
Extended Content (Extended Curriculum Only)
In the Extended curriculum, you must be able to apply the laws of indices to positive, negative, zero, and fractional exponents.
The Rules of Indices
To simplify expressions, apply these rules in order:
- Multiplication Rule: $a^m \times a^n = a^{m+n}$
- Division Rule: $a^m \div a^n = a^{m-n}$
- Power of a Power: $(a^m)^n = a^{mn}$
- Zero Index: $a^0 = 1$ (where $a \neq 0$)
- Negative Indices: $a^{-n} = \frac{1}{a^n}$
- Fractional Indices (Unit Fraction): $a^{\frac{1}{n}} = \sqrt[n]{a}$
- Fractional Indices (General): $a^{\frac{m}{n}} = (\sqrt[n]{a})^m$ or $\sqrt[n]{a^m}$
Worked Example 1: Negative and Zero Indices
Evaluate: $3^{-3} + 4^0$
- Step 1: Apply the negative index rule to $3^{-3}$. $3^{-3} = \frac{1}{3^3}$ Reason: A negative index indicates the reciprocal.
- Step 2: Calculate the value of the denominator. $\frac{1}{3^3} = \frac{1}{27}$ Reason: $3^3 = 3 \times 3 \times 3 = 27$
- Step 3: Apply the zero index rule to $4^0$. $4^0 = 1$ Reason: Any non-zero number raised to the power of 0 is 1.
- Step 4: Add the results together. $\frac{1}{27} + 1 = \frac{1}{27} + \frac{27}{27} = \frac{28}{27}$ Reason: Express 1 as a fraction with a denominator of 27 to add.
Final Answer: $\boxed{\frac{28}{27}}$
Worked Example 2: Fractional Indices
Evaluate: $64^{\frac{5}{6}}$
- Step 1: Identify the root (denominator) and the power (numerator). The root is 6 (6th root), the power is 5. Reason: The denominator of the fractional index indicates the root, and the numerator indicates the power.
- Step 2: Take the 6th root of the base. $\sqrt[6]{64} = 2$ Reason: $2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64$
- Step 3: Raise the result to the power of 5. $2^5 = 32$ Reason: $2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$
Final Answer: $\boxed{32}$
Worked Example 3: Algebraic Simplification
Simplify: $\frac{(8a^9)^{\frac{2}{3}}}{4a^{-2}}$
- Step 1: Apply the power $\frac{2}{3}$ to everything inside the bracket. $(8a^9)^{\frac{2}{3}} = 8^{\frac{2}{3}} \cdot a^{9 \times \frac{2}{3}}$ Reason: Apply the power of a power rule.
- Step 2: Evaluate $8^{\frac{2}{3}}$. $8^{\frac{2}{3}} = (\sqrt[3]{8})^2 = 2^2 = 4$ Reason: The cube root of 8 is 2, and 2 squared is 4.
- Step 3: Simplify the exponent of $a$. $a^{9 \times \frac{2}{3}} = a^6$ Reason: Multiply the indices.
- Step 4: Rewrite the expression. $\frac{4a^6}{4a^{-2}}$
- Step 5: Divide the coefficients and subtract the indices (Division Rule). $\frac{4}{4} \cdot a^{6 - (-2)} = 1 \cdot a^{6+2}$ Reason: Divide the coefficients and subtract the powers.
- Step 6: Simplify. $a^8$
Final Answer: $\boxed{a^8}$
Worked Example 4: Combining Multiple Index Laws
Simplify: $\frac{x^5 \times (x^2)^3}{x^{-2}}$
- Step 1: Simplify the power of a power. $(x^2)^3 = x^{2 \times 3} = x^6$ Reason: Apply the power of a power rule.
- Step 2: Rewrite the expression. $\frac{x^5 \times x^6}{x^{-2}}$
- Step 3: Apply the multiplication rule to the numerator. $x^5 \times x^6 = x^{5+6} = x^{11}$ Reason: When multiplying terms with the same base, add the exponents.
- Step 4: Rewrite the expression. $\frac{x^{11}}{x^{-2}}$
- Step 5: Apply the division rule. $x^{11} \div x^{-2} = x^{11 - (-2)} = x^{11+2} = x^{13}$ Reason: When dividing terms with the same base, subtract the exponents.
Final Answer: $\boxed{x^{13}}$
Key Equations
| Rule Name | Equation | Notes |
|---|---|---|
| Multiplication | $\mathbf{a^m \times a^n = a^{m+n}}$ | Bases must be identical. |
| Division | $\mathbf{a^m \div a^n = a^{m-n}}$ | Subtract the second index from the first. |
| Power of Power | $\mathbf{(a^m)^n = a^{mn}}$ | Multiply the indices. |
| Zero Power | $\mathbf{a^0 = 1}$ | Any base (except 0) to the power 0 is 1. |
| Negative Index | $\mathbf{a^{-n} = \frac{1}{a^n}}$ | Flip the base to make the index positive. |
| Fractional Index | $\mathbf{a^{\frac{m}{n}} = \sqrt[n]{a^m}}$ | "Power over Root". |
Note on Formula Sheet: These rules are not provided on the IGCSE formula sheet. They must be memorised.
Common Mistakes to Avoid
- ❌ Wrong: $3^2 + 3^3 = 3^5$ (Adding indices when adding terms) ✓ Right: $3^2 + 3^3 = 9 + 27 = 36$ (Evaluate each term separately before adding)
- ❌ Wrong: $\frac{5^6}{5^2} = 5^3$ (Incorrect subtraction of indices) ✓ Right: $\frac{5^6}{5^2} = 5^{6-2} = 5^4$ (Subtract the indices correctly)
- ❌ Wrong: $9^{\frac{1}{2}} = 4.5$ (Dividing the base by the denominator of the fractional index) ✓ Right: $9^{\frac{1}{2}} = \sqrt{9} = 3$ (Fractional index of $\frac{1}{2}$ means square root)
- ❌ Wrong: $(a+b)^2 = a^2 + b^2$ (Incorrect expansion of a binomial squared - not directly indices, but related to powers) ✓ Right: $(a+b)^2 = (a+b)(a+b) = a^2 + 2ab + b^2$ (Expand the brackets correctly)
- ❌ Wrong: $4x^0 = 0$ (Incorrect application of the zero index rule) ✓ Right: $4x^0 = 4 \cdot 1 = 4$ (Only $x$ is raised to the power of 0, not the coefficient 4)
Exam Tips
- Command Words:
- "Simplify": Leave your answer in index form (e.g., $x^5$).
- "Evaluate" or "Find the value of": Give the final numerical answer (e.g., $25$).
- Calculator vs. Non-Calculator: This topic often appears on non-calculator papers to test your knowledge of square and cube roots. Memorise squares up to $15^2$ and cubes up to $5^3$ to save time.
- Negative Fractions: If you see a fraction raised to a negative power, e.g., $(\frac{2}{3})^{-2}$, simply "flip" the fraction to make the power positive: $(\frac{3}{2})^2 = \frac{9}{4}$.
- Show Your Working: Even if you use a calculator, write down the intermediate step (e.g., showing that you converted $27^{2/3}$ to $(\sqrt[3]{27})^2$). If you make a small arithmetic error, you can still gain method marks for showing the correct rule.
- Prioritise Roots: When dealing with fractional indices, taking the root before raising to the power often results in smaller, more manageable numbers. For example, in $8^{\frac{4}{3}}$, calculate $\sqrt[3]{8} = 2$ first, then $2^4 = 16$.