Unit Quiz
Overview
Test your mastery of irrational numbers, square roots, and the real number system. This comprehensive quiz covers all unit objectives.
Quiz Questions
Question 1: Simplify sqrt(128).
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Answer: 8sqrt(2)
128 = 64 x 2, so sqrt(128) = sqrt(64) x sqrt(2) = 8sqrt(2).
Question 2: Classify sqrt(49): rational or irrational?
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Answer: Rational
sqrt(49) = 7, which is an integer and therefore rational.
Question 3: Between which two consecutive integers does sqrt(90) lie?
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Answer: Between 9 and 10
81 < 90 < 100, so 9 < sqrt(90) < 10.
Question 4: Simplify 2sqrt(3) + 5sqrt(3).
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Answer: 7sqrt(3)
Like radicals can be combined: 2sqrt(3) + 5sqrt(3) = (2+5)sqrt(3) = 7sqrt(3).
Question 5: Is the sum of a rational and an irrational number always irrational? Explain.
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Answer: Yes
If rational + irrational = rational, then irrational = rational - rational = rational, which is a contradiction.
Question 6: What is the cube root of -27?
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Answer: -3
(-3) x (-3) x (-3) = -27, so the cube root of -27 is -3. Note: cube roots of negative numbers exist.
Question 7: Simplify sqrt(45) + sqrt(20).
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Answer: 5sqrt(5)
sqrt(45) = 3sqrt(5) and sqrt(20) = 2sqrt(5). So 3sqrt(5) + 2sqrt(5) = 5sqrt(5).
Question 8: A square has area 75 cm^2. Express its side length in simplest radical form.
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Answer: 5sqrt(3) cm
Side = sqrt(75) = sqrt(25 x 3) = 5sqrt(3) cm.
Question 9: Order from least to greatest: 2sqrt(5), sqrt(18), 4
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Answer: 4, sqrt(18), 2sqrt(5)
2sqrt(5) is about 4.47, sqrt(18) is about 4.24, and 4 = 4. So: 4 < 4.24 < 4.47.
Question 10: True or False: All real numbers are either rational or irrational, with no overlap.
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Answer: True
By definition, real numbers that can be expressed as a ratio of integers are rational; all others are irrational. These categories are mutually exclusive and exhaustive.