Grade 8 · Grade 8: Real Numbers, Exponents and Scientific Notation · Real Numbers, Exponents and Scale
Rational, Irrational Numbers and Exponent Laws
Sort numbers into rational and irrational, approximate surds on a number line and apply the laws of integer exponents.
Learning objectives
- Distinguish rational from irrational numbers using decimal expansions.
- Approximate irrational numbers between rational bounds.
- Apply the properties of integer exponents.
- Evaluate square roots and cube roots and solve x² = p and x³ = p.
AERO Mathematics alignment
AERO.M8.NS.1
Know that numbers that are not rational are called irrational, and understand that every number has a decimal expansion that is repeating, terminating or non-repeating.
AERO.M8.NS.2
Use rational approximations of irrational numbers to compare their sizes and locate them approximately on a number line.
AERO.M8.NS.3
Know and apply the properties of integer exponents to generate equivalent numerical expressions.
AERO.M8.NS.4
Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, and evaluate square roots of small perfect squares and cube roots of small perfect cubes.
MYP criteria
- Criterion A — Knowing and understanding
- Criterion B — Investigating patterns
- Criterion C — Communicating
ENGAGE
Start here
A square tile has an area of exactly 2 square metres.
Think about it
How long is its side, and can you write that length exactly as a fraction?
Hint: Try to find a fraction whose square is exactly 2.
EXPLAIN
Real numbers, powers and roots
A rational number can be written as a ratio of two integers; its decimal expansion always terminates or repeats. An irrational number cannot, and its decimal expansion never terminates and never repeats. √2 and π are irrational. Together the rationals and irrationals form the real numbers.
Approximating irrational numbers
Trap the value between rational bounds: 1.4² = 1.96 and 1.5² = 2.25, so √2 lies between 1.4 and 1.5. Refine to 1.41 and 1.42 to get another decimal place.
Laws of integer exponents
| Law | Rule | Example |
|---|---|---|
| Product | xᵃ · xᵇ = xᵃ⁺ᵇ | 2³ · 2⁴ = 2⁷ |
| Quotient | xᵃ ÷ xᵇ = xᵃ⁻ᵇ | 5⁶ ÷ 5² = 5⁴ |
| Power of a power | (xᵃ)ᵇ = xᵃᵇ | (3²)⁴ = 3⁸ |
| Zero | x⁰ = 1 (x ≠ 0) | 7⁰ = 1 |
| Negative | x⁻ᵃ = 1/xᵃ | 2⁻³ = 1/8 |
Roots undo powers
If x² = 49 then x = ±7; if x³ = 64 then x = 4. Note that a square equation has two real solutions while a cube equation has one.
Common misconception
An irrational number is not simply a very long decimal. Its digits never fall into any repeating block, which is why no fraction can equal it exactly.
INVESTIGATION
Try it yourself
Use a calculator to square 1.41, 1.414 and 1.4142. Record how close each gets to 2, then explain why no decimal will ever land exactly on 2.
Watch
Laws of Exponents
Math Antics · 8:00
The exponent rules, explained from repeated multiplication rather than memorised.
Before you watch: Why is x³ · x⁴ equal to x⁷ and not x¹²?
- Simplify (2³)² · 2⁻⁴.
- Between which two whole numbers does √50 lie?
Interactive simulation · PhET
Number Line: Distance
Place √2, √3 and π between whole numbers, then refine your bounds one decimal at a time.
While you explore
- Between which two tenths does √7 lie?
- How do you know √7 is irrational?
Key vocabulary
- Cube root
- The number that gives your value when multiplied by itself three times.
- Irrational number
- A number that cannot be written as a fraction.
Practice questions
0/1 correct
Level 1 · Criterion A
Simplify 2⁵ ÷ 2².
MYP criterion tasks
Level 2 · Criterion B
Between which two consecutive tenths does √30 lie? Show your reasoning.
Reflect & track
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