Review: real numbers, intervals, radicals and exponents
Mathematics Prep 2 Term 1, Units 1–4, Review
GoalReview Unit 1: sort and order real numbers, work with intervals, simplify radicals and use the laws of exponents in new mixed tasks.
New words
Q′ · the irrational numbers: R = Q ∪ Q′ and Q ∩ Q′ = ∅conjugate · a + √b and a − √b; their product is a rational numberinterval · a set of real numbers between two ends; a square bracket includes the end, an open one leaves it outa⁰ and a⁻ⁿ · a⁰ = 1 for a ≠ 0, and a⁻ⁿ = 1/aⁿ
Explanation
Key facts of Unit 1. A square root of a number that is not a perfect square, a cube root of a number that is not a perfect cube, and π are irrational; a decimal that stops or repeats is rational. Estimate a root between two whole numbers and refine it with a calculator, and order numbers by comparing their decimals. Intervals: a square bracket means the end is included (≤) and an open bracket means it is left out (<); ∞ always has an open bracket. For union, intersection, difference and complement, draw a number line. For radicals: add like radicals only, √a × √b = √(ab), (√a)² = a, simplify by taking out square factors, and rationalise by multiplying top and bottom by the conjugate. Exponent laws: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ. Traps: (−2)⁴ = 16 but −2⁴ = −16; aᵐ + aⁿ cannot be joined; x² = 7 has no rational solution.
Expression
Fact
√20, ∛80, π, 0.1010010001…
irrational (not repeating)
0.25, 22/5, −√9
rational
[a, b[ , ]a, b]
one end in, one end out
(a + √b)(a − √b)
a² − b
aᵐ × aⁿ
aᵐ⁺ⁿ
a⁻ⁿ
1/aⁿ
Worked examples
Sort and order. Which of √20, 4.5, 22/5, ∛80, −√9 are rational? 4.5, 22/5 = 4.4 and −√9 = −3 are rational; √20 ≈ 4.47 and ∛80 ≈ 4.31 are irrational. In order: −3 < ∛80 < 22/5 < √20 < 4.5, because 4.31 < 4.4 < 4.47 < 4.5.
Intervals. A = [−3, 5[ and B = ]1, 8]. Then A ∩ B = ]1, 5[, A ∪ B = [−3, 8], A − B = [−3, 1], B − A = [5, 8], and the complement of A in R is ]−∞, −3[ ∪ [5, ∞[. Check the ends: 1 is in A but not in B, so it belongs to A − B; 5 is in B but not in A.
Make a quiz. In pairs, write five questions about Unit 1 (one on each: irrational numbers, intervals, radicals, conjugates, exponents). Swap and solve, then mark each other’s answers.
Practice
1
Simplify √48 + √27 − 2√3.
4√3 + 3√3 − 2√3 = 5√3.
2
Write ]−2, 6] ∩ [4, 9[ as an interval, and list the integers in it.
[4, 6]; the integers are 4, 5, 6.
3
Rationalise 6 ÷ (3 − √3).
6(3 + √3) ÷ (9 − 3) = 3 + √3.
4
Simplify x⁻² × x⁵ ÷ x² and find its value for x = 7.
x⁻²⁺⁵⁻² = x¹ = 7.
5
Is √2 + (−√2) rational? Is √2 × √2 rational?
Yes: the sum is 0 and the product is 2. A sum or product of irrationals can be rational.
