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75 multiple-choice questions and 21 flashcards on Approximating Irrational Numbers on the Number Line, about 50% of the 8th Math: The Number System bank. Every one carries a written rationale.
Approximating Irrational Numbers on the Number Line is one of 2 chapters in CoStudy's 8th Math: The Number System bank, and it holds 75 of the bank's 150 multiple-choice questions — roughly 50% of the total. That proportion is not arbitrary: chapters follow the certifying body's published exam outline, and the number of questions in each is set by that domain's published weight, so the share of your practice time this chapter takes matches the share of the real exam it accounts for.
Studying by chapter is worth doing once you have a diagnostic score. A single overall percentage tells you whether you are close; it does not tell you which domain is dragging. Working a weak chapter in isolation, and re-testing it in isolation, is the fastest way to move a score that has stalled — and it is why the mock exams in CoStudy report by domain rather than as one number.
10 questions drawn from this chapter, with the full rationale shown — the controlling principle behind the right answer, and why each wrong option tempts and fails.
Which is between √3 and √5?
Answer: B — 2
B) √3 ≈ 1.73 and √5 ≈ 2.24; 2 is between them. A) 1.5 < √3. C) 2.5 > √5. D) Way above.
√(−16) is:
Answer: C — Not a real number
C) Square roots of negative numbers are not real. A) (−4)² = +16, not −16. B) Same. D) i isn't covered in Grade 8 real-number context.
Which correctly expresses 0.00045 in scientific notation?
Answer: A — 4.5 × 10⁻⁴
Move decimal 4 places right to get 4.5 → 4.5 × 10⁻⁴. (B) coefficient must be 1 ≤ |a| < 10; (C) positive exponent means larger number; (D) coefficient must be ≥ 1.
Which is closest to √80?
Answer: B — 8.9
B) 8² = 64, 9² = 81; √80 ≈ 8.94. A) That's √64. C) √81. D) Half of 80.
Order from least to greatest: 3, π, √8, 22/7.
Answer: A — √8, 3, π, 22/7
A) √8 ≈ 2.83 < 3 < π ≈ 3.14159 < 22/7 ≈ 3.14286. B) Puts √8 in wrong spot (it's less than 3). C) Has 22/7 < π — wrong, they're very close but 22/7 > π. D) Same error placing √8.
For positive x, if x² = 13, then x is:
Answer: C — Between 3 and 4
C) 3² = 9, 4² = 16; 9 < 13 < 16, so √13 ≈ 3.6. A) 3.5² = 12.25 ≠ 13. B) Off-by-one error using 4² and 5². D) 13 isn't a perfect square.
x³ = −64. What is x?
Answer: A — −4
(−4)³ = −64, so x = −4. Cube roots of negative numbers are real and negative. (B) 4³ = +64; (C) −8 cubed = −512; (D) cube roots of negatives always exist.
Which is the BEST estimate of √40?
Answer: A — 6.3
A) 6² = 36, 7² = 49; √40 ≈ 6.32. B) That's √30. C) That's √49. D) Half of 40 — wrong operation.
Which value is closest to √2 + √3?
Answer: B — 3.1
√2 ≈ 1.414; √3 ≈ 1.732; sum ≈ 3.146. Closest to 3.1 (B). (A) too small; (C) 3.5 > 3.146; (D) too large.
√(9/16) =
Answer: A — 3/4
√(9/16) = √9/√16 = 3/4. (B) didn't take the root; (C) took root of numerator but not denominator; (D) same mistake.
4 cards from the 21 in this chapter.
√50 simplified?
5√2.
What is √2 approximately?
≈ 1.414.
π approximately equals?
3.14159... (irrational).
Real numbers include?
All rational and irrational numbers.
These are a sample. The full Approximating Irrational Numbers on the Number Line chapter runs 96 items with per-chapter progress tracking, on the web and in the iOS app.
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