Indices I
Section: Number 1 | Syllabus: Cambridge IGCSE Mathematics (0580)
Positive, Zero and Negative Indices
An index (or power) tells you how many times a number is multiplied by itself. Extending the pattern of indices below zero, rather than stopping at 1, is what gives meaning to a zero index and a negative index.
- an means a multiplied by itself n times, where n is a positive integer
- Any nonzero number raised to the power 0 equals 1: a^0 = 1
- A negative index means the reciprocal of the positive power: a^-n = (1/a^n)
| 5³ | 5² | 5¹ | 5⁰ | 5⁻¹ | 5⁻² |
|---|---|---|---|---|---|
| 125 | 25 | 5 | 1 | 1/5 | 1/25 |
Each step down divides by 5 - the pattern carries on smoothly through the power of 0 and into negative powers
Worked Example: Evaluating Positive, Zero and Negative Indices
- 3^4 = 3 × 3 × 3 × 3 = 81
- 9^0 = 1
- 2^-5 = (1/2^5) = (1/32)
- 10^-1 = (1/10) = 0.1
Common Mistakes
MistakeBelieving that a number raised to the power 0 equals 0, e.g. writing 9^0 = 0
Fixany nonzero number raised to the power 0 is always 1, not 0
MistakeTreating a negative index as making the value negative, e.g. writing 2^-5 = -32
Fixa negative index means "take the reciprocal", not "make negative": 2^-5 = (1/2^5) = (1/32)
The Laws of Indices
When powers with the same base are combined, a small set of laws let you rewrite the result as a single power without expanding anything. Both letters and numbers work with the same laws.
- Any number to the power 1 is itself: a^1 = a
- To multiply powers with the same base, add the indices: a^m × a^n = a^m+n
- To divide powers with the same base, subtract the indices: a^m ÷ a^n = a^m-n
- To raise a power to a further power, multiply the indices: (a^m)^n = a^mn
- To raise a product to a power, apply the power to each factor separately: (ab)^n = a^n b^n
- To raise a fraction to a power, apply the power to the numerator and denominator separately: ((a/b))^n = (a^n/b^n)
When the bases match, multiplying powers means adding their indices
Worked Example: Why the Laws Work
- Multiplying: 2^4 × 2^3 = (2222) × (222) = 2^7 - seven 2s multiplied together, so the indices simply add
- Dividing: 5^5 ÷ 5^3 = (55555/555) = 55 = 5^2 - three of the 5s on top cancel with the three on the bottom, leaving two
- Power of a power: (3^2)^4 = (33)×(33)×(33)×(33) = 3^8 - four groups of two 3s is eight 3s in total
- Power of a product: (25)^3 = (25)×(25)×(25) = (222)×(555) = 2^3 × 5^3 - the factors can be regrouped and raised to the power separately
- Power of a fraction: ((3/4))^2 = (3/4)×(3/4) = (33/44) = (3^2/4^2) = (9/16)
Worked Example: Simplifying with the Laws of Indices
- Question: Simplify (6^2 × 6^5) ÷ 6^4, giving your answer as a single power of 6.
- Step 1: Multiply the powers in the brackets by adding indices: 6^2 × 6^5 = 6^(2+5) = 6^7
- Step 2: Divide by 6^4 by subtracting indices: 6^7 ÷ 6^4 = 6^(7-4) = 6^3
- Answer: 6^3
Worked Example: Writing a Product as a Single Power
- Write 4 × 32 as a single power of 2: 4 = 2^2,\ 32 = 2^5,\ so 4 × 32 = 2^2 × 2^5 = 2^7
- Write 25 ÷ 125 as a single power of 5: 25 = 5^2,\ 125 = 5^3,\ so 25 ÷ 125 = 5^2 ÷ 5^3 = 5^-1
Worked Example: Applying a Power to a Product or Fraction
- Question: Simplify (2x^3)^4.
- Step 1: Apply the power 4 to each factor separately: 2^4 × (x^3)^4
- Step 2: Evaluate 2^4 = 16, and simplify (x^3)^4 = x^12 using the power-of-a-power rule
- Answer: 16x^12
Common Mistakes
MistakeMultiplying the indices when the powers themselves are being multiplied, e.g. writing 6^2 × 6^5 = 6^10
Fixadd the indices when multiplying powers, and only multiply the indices when raising a power to a further power: 6^2 × 6^5 = 6^7,\ but (6^2)^5 = 6^10
MistakeSubtracting the indices in the wrong order when dividing, e.g. writing 6^4 ÷ 6^7 = 6^3
Fixalways subtract the second index from the first, in order: 6^4 ÷ 6^7 = 6^(4-7) = 6^-3
MistakeApplying the power to only one factor of a product, e.g. writing (2x^3)^4 = 2x^12
Fixthe power applies to every factor inside the brackets: (2x^3)^4 = 2^4 × (x^3)^4 = 16x^12
Fractional Indices
Extended Only
A fractional index combines a root with a power. Once a fractional index is linked to a root, the same laws of indices from above still apply to it.
- A power of 1/n means the nth root: a^(1/n) = √[n]a
- A power of m/n means take the nth root, then raise the result to the power m: a^(m/n) = (√[n]a)^m
- A negative fractional index combines both ideas: work out the root and power first, then take the reciprocal
Worked Example: Evaluating a Fractional Index
- Question: Evaluate 4^-(3/2).
- Step 1: Deal with the root first: 4^(1/2) = √(4) = 2
- Step 2: Raise the result to the power 3: 2^3 = 8
- Step 3: The index was negative, so take the reciprocal: 4^-(3/2) = (1/8)
- Answer: 1/8
Worked Example: More Fractional Indices
- 27^(1/3) = √[3]27 = 3
- 32^(1/5) = √[5]32 = 2
- 9^(3/2) = (√(9))^3 = 3^3 = 27
Common Mistakes
MistakeTreating a fractional index as ordinary division, e.g. thinking 27^(1/3) means 27 ÷ 3 = 9
Fixa fractional index of 1/n means the nth root, not division by n: 27^(1/3) = √[3]27 = 3
MistakeForgetting the final reciprocal step when the fractional index is negative, e.g. leaving 4^-(3/2) as 8
Fixwork out the root and power first, then take the reciprocal because the index is negative: 4^-(3/2) = (1/8)
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