NCEA Level 2Indices and surdsAS 91261

Index laws, fractional indices and surds

What this lesson covers

  • Index laws apply to every factor inside a bracket, so a power of a product raises each part.
  • A surd can be written as a fractional index, which lets the index laws do the work.
Key idea

Indices are shorthand

  • y³ means y × y × y.
  • Every index law follows from that.
  • When unsure, write it out.
Key idea

The index laws

  • Multiply: add the powers.
  • Divide: subtract them.
  • Power of a power: multiply them.
LawFormResult
Multiplyaᵐ × aⁿaᵐ⁺ⁿ
Divideaᵐ ÷ aⁿaᵐ⁻ⁿ
Power of a power(aᵐ)ⁿaᵐⁿ
Zeroa⁰1
Negativea⁻ⁿ1/aⁿ
Worked example

(2y)³ is not 2y³

  • Write it out: 2y × 2y × 2y.
  • The numbers: 2 × 2 × 2 = 8.
  • The letters: y × y × y = y³. So 8y³.
(2y)³ is not 2y³
Worked example

Every factor inside gets the power

  • Each factor is raised.
  • Including a minus sign.
  1. (3a²b)² = 3² × (a²)² × b²every factor
  2. = 9 × a⁴ × b²multiply the powers
  3. = 9a⁴b²
  4. (−2x)² = (−2)² × x²the minus too
  5. = 4x²(−2)² = 4
Worked example

Negative indices mean one over

  • A negative index: one over.
  • Check what is raised.
  1. x⁻³ = 1/x³one over
  2. 2x⁻³ = 2 × 1/x³ = 2/x³only x is raised
  3. (2x)⁻¹ = 1/(2x)the whole bracket
  4. x⁻² ÷ x⁻⁵ = x^(−2 − (−5))subtract
  5. = x³
Worked example

Fractional indices are roots

  • Denominator: the root.
  • Numerator: the power.
  1. 8^(2/3) = (∛8)²root, then power
  2. = 2² = 4∛8 = 2
  3. 27^(−1/3) = 1/27^(1/3)negative: one over
  4. = 1/∛27 = 1/3∛27 = 3
Key idea

Why √x is x^(1/2)

  • Add the indices.
  • It squares to give x.
  1. x^(1/2) × x^(1/2)
  2. = x^(1/2 + 1/2)add the indices
  3. = x¹ = x
  4. so x^(1/2) = √x
Worked example

Surd form to index form

  • A root is a power.
  • Then use the index laws.
  1. √(x⁵) = (x⁵)^(1/2)a root is power 1/2
  2. = x^(5 × 1/2) = x^(5/2)multiply indices
  3. 1/√x = 1/x^(1/2)
  4. = x^(−1/2)one over: negative
Worked example

Simplifying surds

  • Take out square factors.
  • Like surds add.
  • No surd underneath.
  1. √50 = √25 × √2 = 5√2square factor 25
  2. √12 = √4 × √3 = 2√3
  3. √27 = √9 × √3 = 3√3
  4. √12 + √27 = 2√3 + 3√3like surds
  5. = 5√3
  6. 6/√3 × √3/√3rationalise
  7. = 6√3/3
  8. = 2√3
Key idea

Where the marks are

  • Achieved: simplify and convert accurately.
  • Merit: combine several laws in one expression.
  • Simplest form, every time.
Summary

What to take away

  • Every factor gets the power. Roots are fractions.
Bracketsthe power applies to every factor
Negative indexone over
Fractional indexdenominator is the root
Surd to index√x = x^(1/2)
Simplify surdstake out square factors
Rationalisemultiply top and bottom by the surd
Your turn

Your turn: four questions

  • Pause here and answer each one.
  • No calculator for question 4.
  • Simplest form.
Question
1Simplify (4x³)².
2Simplify 12a⁵b² ÷ 3a²b⁵.
3Write √(x³) ÷ x as a single power of x.
4Evaluate 16^(3/4) without a calculator.
Show answer

1. 16x⁶: both the 4 and the x³ are squared. 2. 4a³b⁻³, which is 4a³/b³. 3. x^(3/2) ÷ x¹ = x^(1/2). 4. The fourth root of 16 is 2, and 2³ = 8.

Worked example

The answers: questions 1 and 2

  1. 1. (4x³)² = 4² × (x³)²every factor
  2. = 16x⁶(x³)² = x⁶
  3. 2. 12a⁵b² ÷ 3a²b⁵
  4. = (12 ÷ 3) a⁵⁻² b²⁻⁵subtract the powers
  5. = 4a³b⁻³
  6. = 4a³/b³negative: one over
Worked example

The answers: questions 3 and 4

  1. 3. √(x³) ÷ x = x^(3/2) ÷ x¹surd to index
  2. = x^(3/2 − 1)subtract the powers
  3. = x^(1/2)
  4. 4. 16^(3/4) = (⁴√16)³root, then power
  5. = 2³ = 8⁴√16 = 2

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