Indices I

1. Overview

Indices (also known as powers or exponents) are a way to express repeated multiplication concisely. Understanding and applying the laws of indices is crucial for simplifying algebraic expressions, solving equations, and working with standard form, logarithms, and more advanced mathematical concepts. This revision note covers the core concepts and rules of indices required for the IGCSE Cambridge Mathematics (0580) syllabus, including positive, zero, negative, and fractional indices.

Key Definitions

• Base: The number or variable being multiplied by itself (e.g., in $5^3$, 5 is the base).
• Index (plural Indices): The number indicating how many times to multiply the base by itself (e.g., in $5^3$, 3 is the index). Also called the power or exponent.
• Reciprocal: The result of dividing 1 by a number (e.g., the reciprocal of $x$ is $\frac{1}{x}$).
• Root: The inverse operation of an index (e.g., the square root $\sqrt{x}$ is the inverse of $x^2$).

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Core Content

The Laws of Indices

These rules apply only when the bases are the same.

1. Multiplication Rule: $\mathbf{a^m \times a^n = a^{m+n}}$ To multiply terms with the same base, add the indices.

• Algebraic Example: $x^5 \times x^3 = x^{(5+3)} = x^8$
• Numerical Example:
  • $2^3 \times 2^2$
  • $= (2 \times 2 \times 2) \times (2 \times 2)$
  • $= 2^5 = 32$

2. Division Rule: $\mathbf{a^m \div a^n = a^{m-n}}$ To divide terms with the same base, subtract the indices.

• Algebraic Example: $y^9 \div y^4 = y^{(9-4)} = y^5$
• Numerical Example:
  • $5^6 \div 5^4$
  • $= 5^{(6-4)}$
  • $= 5^2 = 25$

3. Power of a Power Rule: $\mathbf{(a^m)^n = a^{m \times n}}$ To raise a power to another power, multiply the indices.

• Algebraic Example: $(z^4)^3 = z^{(4 \times 3)} = z^{12}$
• Numerical Example:
  • $(3^2)^2$
  • $= 3^{(2 \times 2)}$
  • $= 3^4 = 81$

Zero and Negative Indices

1. Zero Index: $\mathbf{a^0 = 1}$ Any non-zero number raised to the power of zero is exactly 1.

• Example: $100^0 = 1$; $(xyz)^0 = 1$.

2. Negative Indices: $\mathbf{a^{-n} = \frac{1}{a^n}}$ A negative index indicates a reciprocal.

• Algebraic Example: $x^{-3} = \frac{1}{x^3}$
• Numerical Example:
  • Evaluate $4^{-2}$
  • $= \frac{1}{4^2}$
  • $= \frac{1}{16}$

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Extended Content (Extended Curriculum Only)

Fractional Indices

Fractional indices represent roots and powers of a number.

1. Unit Fractions: $\mathbf{a^{\frac{1}{n}} = \sqrt[n]{a}}$ The denominator of the fraction indicates the type of root to take.

• Example: $144^{\frac{1}{2}} = \sqrt{144} = 12$
• Example: $27^{\frac{1}{3}} = \sqrt[3]{27} = 3$

2. General Fractions: $\mathbf{a^{\frac{m}{n}} = (\sqrt[n]{a})^m}$ The denominator ($n$) is the root, and the numerator ($m$) is the power. It is generally easier to take the root first, then apply the power.

Worked example 1 — Evaluate a fractional index

Question: Evaluate $8^{\frac{2}{3}}$

1. $8^{\frac{2}{3}}$
   • Original expression
2. $= (\sqrt[3]{8})^2$
   • Rewrite as a root and a power
3. $= (2)^2$
   • Evaluate the cube root of 8
4. $= 4$
   • Square the result

Final Answer: $\mathbf{4}$

Worked example 2 — Simplify an expression with a fractional index

Question: Simplify $(16x^8)^{\frac{3}{4}}$

1. $(16x^8)^{\frac{3}{4}}$
   • Original expression
2. $= 16^{\frac{3}{4}} \times (x^8)^{\frac{3}{4}}$
   • Apply the power to both the coefficient and the variable
3. $= (\sqrt[4]{16})^3 \times x^{8 \times \frac{3}{4}}$
   • Rewrite the coefficient as a root and a power, and multiply the exponents of the variable
4. $= (2)^3 \times x^6$
   • Evaluate the fourth root of 16 and simplify the exponent of x
5. $= 8x^6$
   • Cube the result

Final Answer: $\mathbf{8x^6}$

Worked example 3 — Combining index laws with fractional indices

Question: Simplify $\frac{9^{\frac{3}{2}} \times 3^{-1}}{3^2}$

1. $\frac{9^{\frac{3}{2}} \times 3^{-1}}{3^2}$
   • Original expression
2. $= \frac{(3^2)^{\frac{3}{2}} \times 3^{-1}}{3^2}$
   • Rewrite 9 as $3^2$
3. $= \frac{3^{2 \times \frac{3}{2}} \times 3^{-1}}{3^2}$
   • Apply the power of a power rule
4. $= \frac{3^3 \times 3^{-1}}{3^2}$
   • Simplify the exponent
5. $= \frac{3^{3 + (-1)}}{3^2}$
   • Apply the multiplication rule
6. $= \frac{3^2}{3^2}$
   • Simplify the exponent
7. $= 3^{2-2}$
   • Apply the division rule
8. $= 3^0$
   • Simplify the exponent
9. $= 1$
   • Apply the zero index rule

Final Answer: $\mathbf{1}$

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Key Equations

Rule	Equation	Note
Multiplication	$\mathbf{a^m \times a^n = a^{m+n}}$	Bases must be identical
Division	$\mathbf{a^m \div a^n = a^{m-n}}$	Bases must be identical
Power of Power	$\mathbf{(a^m)^n = a^{mn}}$	Multiply the indices
Zero Index	$\mathbf{a^0 = 1}$	$a \neq 0$
Negative Index	$\mathbf{a^{-n} = \frac{1}{a^n}}$	Flip to make positive
Fractional Index	$\mathbf{a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m}$	Root first, then power

Note: These formulas are NOT provided on the IGCSE formula sheet. You must memorise them.

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Common Mistakes to Avoid

• ❌ Wrong: $x^2 \times x^3 = x^6$ ✓ Right: $x^2 \times x^3 = x^{2+3} = x^5$. (When multiplying terms with the same base, add the indices, not multiply them).
• ❌ Wrong: $(2x)^3 = 2x^3$ ✓ Right: $(2x)^3 = 2^3 \times x^3 = 8x^3$. (The power applies to both the coefficient and the variable).
• ❌ Wrong: $9^{\frac{1}{2}} = \frac{1}{3}$ ✓ Right: $9^{\frac{1}{2}} = \sqrt{9} = 3$. (A fractional index of $\frac{1}{2}$ represents the square root, not the reciprocal).
• ❌ Wrong: $4^{-1} = -4$ ✓ Right: $4^{-1} = \frac{1}{4}$. (A negative index indicates a reciprocal, not a negative number).
• ❌ Wrong: Simplifying $(x^4)^2$ as $x^6$ ✓ Right: $(x^4)^2 = x^{4 \times 2} = x^8$. Remember to multiply the indices when raising a power to another power.

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Exam Tips

• Command Words:
  • "Simplify": Leave your answer in index form (e.g., $3^7$).
  • "Evaluate" or "Calculate": Find the final numerical value (e.g., $27$).
• Calculator vs Non-Calculator:
  • In non-calculator sections, you are expected to know the powers of 2 (up to $2^6$), 3 (up to $3^4$), and 5 (up to $5^3$).
  • If you see a large number (like 32 or 81), try rewriting it as a power of a prime number ($32 = 2^5$, $81 = 3^4$) to simplify the expression.
• Showing Work: You will lose marks if you jump straight to an answer on multi-step fractional index questions. Always show the root step and the power step separately.
• Typical Values: Be very comfortable with $x^{\frac{1}{2}}$ being $\sqrt{x}$ and $x^{-1}$ being $\frac{1}{x}$. These appear frequently in algebraic rearrangement questions.