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Irrational Numbers Test Answer Key with Scoring – Grade 8

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Last updated: October 22, 2026 · Written by the Math Made Easy math teachers

Grade 8 Math Test Answer Keys – Mathbot, the Math Made Easy robot coach, with free printable test answer keys for Grade 8
Looking for an irrational numbers test answer key? Then this answer key gives the solutions and the scoring guide of the Grade 8 test on Rational and Irrational Numbers Explained. Use this answer key to score each question. Partial credit is listed in the scoring line under every solution. Award points for correct reasoning, such as testing squares, even when the arithmetic slips at the end.

Standards: 8.NS.A.1, 8.NS.A.2 · Level: Grade 8

Question 1 Sort the numbers (3 points)

  1. −5.25 = −214: rational.
  2. √72 is between 8 and 9 and not a perfect square: irrational.
  3. √169 = 13: rational.
  4. Repeating block: rational.
  5. 2π: irrational.
  6. √916 = 34: rational.

Irrational: √72 and 2π. Rational: the other four.

Scoring: ½ point per correct label.

Question 2 Decimals and fractions (3 points)

  1. 0.6125 = 6,12510,000 = 4980.
  2. Let x = 0.135135…; 1,000x = 135.135…; 999x = 135; x = 135999 = 537.

(a) 4980 (b) 537

Scoring: (a) 1 point. (b) 1 point for the equation 999x = 135, 1 point for the simplified fraction.

Question 3 Estimating a root (2 points)

7² = 49 and 8² = 64, so 7 < √61 < 8. Next, 7.8² = 60.84 and 7.9² = 62.41. Since 61 is much closer to 60.84, the estimate is 7.8.

Between 7 and 8; √61 ≈ 7.8

Scoring: 1 point for the integers, 1 point for 7.8 with a supporting square.

Question 4 Points on a number line (3 points)

3.3² = 10.89 and 3.4² = 11.56, so √11 ≈ 3.32: point B. 3.6² = 12.96 and 3.7² = 13.69, so √13 ≈ 3.61: point C. 3.8² = 14.44 and 3.9² = 15.21, so √15 ≈ 3.87: point D. Point A is 3.1, which is not one of the three roots.

3.03.54.0AB = √11C = √13D = √15
Matched points.

B = √11, C = √13, D = √15

Scoring: 1 point per correct match, with its square check.

Question 5 Order the numbers (2 points)

479 = 4.777… . 4.79² = 22.9441, less than 23, so 4.79 < √23. 4.8² = 23.04, more than 23, so √23 < 4.8. Order: 4.777… < 4.79 < √23 < 4.8.

479, 4.79, √23, 4.8

Scoring: 1 point for the correct order, 1 point for squaring to justify √23.

Question 6 The mural (3 points)

  1. 9² = 81 and 10² = 100, so the side is between 9 and 10 feet.
  2. 9.4² = 88.36 and 9.5² = 90.25. Since 90 is closer to 90.25, the side is about 9.5.
  3. No. 9.5² = 90.25, not 90, so 9.5 is only an approximation of √90, which is irrational.

(a) 9 and 10 (b) 9.5 ft (c) No, 9.5² = 90.25

Scoring: 1 point per part.

Question 7 Always irrational? (2 points)

Take √2 and √8, both irrational. √2 · √8 = √16 = 4, and 4 is rational. Another example is √3 · √3 = 3.

Claim false: √2 · √8 = 4.

Scoring: 1 point for two irrational factors, 1 point for the rational product.

Question 8 A late-starting repeat (2 points)

Let x = 0.2777…. Then 10x = 2.777… and 100x = 27.777…. Subtract: 90x = 25, so x = 2590 = 518.

518

Scoring: 1 point for 90x = 25, 1 point for the simplified fraction.

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