Year 8 Number Cheatsheet

• Index notation: a number raised to a power (exponent) shows how many times to multiply it
• Exponent laws help us simplify expressions with powers
• Rational numbers include fractions, decimals, and integers — any number we can write as a ratio
• Irrational numbers cannot be written as a simple fraction (like π)
• Percentages, rates, and ratios compare quantities in real-world contexts
• Terminating decimals end (like 0.5), recurring decimals repeat forever (like 0.333...)

Example 1: Simplify 2⁴ × 2³ ÷ 2²

1. Use product of powers: 2⁴ × 2³ = 2⁴⁺³ = 2⁷
2. Now divide: 2⁷ ÷ 2² = 2^7-2 = 2⁵
3. Calculate: 2⁵ = 32

Example 2: A shirt costs $40. It is reduced by 15%. What is the new price?

1. Calculate the reduction: 15% of $40 = 0.15 × 40 = $6
2. Subtract from original: $40 - $6 = $34
3. Alternatively: New price = 40 × (1 - 0.15) = 40 × 0.85 = $34

Example 3: A map has a ratio of 1:50,000. If a distance on the map is 4 cm, what is the actual distance in km?

1. The ratio 1:50,000 means 1 cm on map = 50,000 cm in reality
2. Actual distance = 4 × 50,000 = 200,000 cm
3. Convert to km: 200,000 cm ÷ 100,000 = 2 km

• Mistake: 2³ × 2² = 4⁵ (multiplying bases as well as adding exponents)
  Correction: Only add exponents when bases are the same: 2³ × 2² = 2⁵ = 32
• Mistake: 0.5 is irrational because it has a decimal point
  Correction: 0.5 = 1/2, which is rational. It's a terminating decimal!
• Mistake: To find a 20% increase, multiply by 20 (instead of 1.20)
  Correction: A 20% increase means multiply by 1 + 0.20 = 1.20
• Mistake: 3:5 means 3 is bigger than 5
  Correction: A ratio just shows the relationship; 3:5 means for every 3 of item A, there are 5 of item B

1. Simplify: 5⁶ ÷ 5⁴
   Answer

   5^6-4 = 5² = 25
2. A video game costs $60 and is on sale for 25% off. What is the sale price?
   Answer

   $60 × (1 - 0.25) = $60 × 0.75 = $45
3. Convert 0.666... to a fraction
   Answer

   This recurring decimal equals 2/3
4. A recipe uses flour and sugar in the ratio 3:2. If you use 9 cups of flour, how much sugar is needed?
   Answer

   If flour is 3 parts and you use 9 cups, then 1 part = 3 cups. Sugar (2 parts) = 2 × 3 = 6 cups
5. Evaluate (3²)³
   Answer

   (3²)³ = 3^2×3 = 3⁶ = 729

• Exponent (or Power, Index): The small number that tells you how many times to multiply. In 2³, the exponent is 3.
• Base: The number being multiplied. In 2³, the base is 2.
• Rational Number: Any number that can be written as a fraction a/b (where a and b are integers).
• Irrational Number: A number that cannot be written as a simple fraction, like π or √2.
• Terminating Decimal: A decimal that stops, like 0.25 or 1.5.
• Recurring Decimal: A decimal where digits repeat forever, like 0.333... or 0.142857142857...
• Ratio: A comparison between two quantities, written as a:b or a/b.
• Percentage: A way of expressing a number as a fraction of 100.