Common Mistakes
Overview
Recognize and avoid common errors when working with irrational numbers, square roots, and the real number system. Understanding these pitfalls will improve your accuracy.
Practice Problems
Question 1: A student says sqrt(9 + 16) = sqrt(9) + sqrt(16) = 3 + 4 = 7. What's the error?
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Answer: You cannot distribute a square root over addition
Correct: sqrt(9 + 16) = sqrt(25) = 5, not 7. The property sqrt(a) + sqrt(b) does NOT equal sqrt(a + b).
Question 2: A student claims 0.333... (repeating) is irrational because the decimal never ends. Is this correct?
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Answer: No - 0.333... is rational
Repeating decimals are always rational. 0.333... = 1/3. Only non-repeating, non-terminating decimals are irrational.
Question 3: A student simplifies sqrt(50) as sqrt(25) x sqrt(2) = 25sqrt(2). Find the error.
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Answer: Should take the square root of 25
sqrt(50) = sqrt(25 x 2) = sqrt(25) x sqrt(2) = 5sqrt(2), not 25sqrt(2).
Question 4: A student says pi = 3.14, so pi is rational. What's wrong with this reasoning?
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Answer: 3.14 is just an approximation, not the exact value
Pi is irrational - it cannot be exactly expressed as a fraction or terminating/repeating decimal. 3.14 is a rounded approximation.
Question 5: A student writes sqrt(x^2) = x. When is this incorrect?
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Answer: When x is negative
sqrt(x^2) = |x| (absolute value), not x. For example, sqrt((-3)^2) = sqrt(9) = 3, not -3.
Question 6: A student estimates sqrt(150) is between 10 and 20 because 10^2 = 100 and 20^2 = 400. Can they give a better estimate?
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Answer: Yes - sqrt(150) is between 12 and 13
12^2 = 144 and 13^2 = 169, so 12 < sqrt(150) < 13. The estimate can be narrowed to about 12.2.
Question 7: A student says all integers are rational numbers. Is this correct?
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Answer: Yes, this is correct
Every integer n can be written as n/1, which is a ratio of integers. So all integers are rational.
Question 8: A student adds 3sqrt(2) + 4sqrt(3) and gets 7sqrt(5). What's wrong?
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Answer: Cannot combine unlike radicals
3sqrt(2) + 4sqrt(3) cannot be simplified because the radicands (2 and 3) are different. The answer stays as 3sqrt(2) + 4sqrt(3).
Question 9: A student says the product of two irrational numbers is always irrational. Give a counterexample.
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Answer: sqrt(2) x sqrt(2) = 2 (rational)
The product of two irrationals can be rational. Another example: sqrt(8) x sqrt(2) = sqrt(16) = 4.
Question 10: A student concludes sqrt(4/9) = sqrt(4)/sqrt(9) = 2/3. Is this correct?
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Answer: Yes, this is correct
Unlike addition, you CAN distribute square roots over division: sqrt(a/b) = sqrt(a)/sqrt(b). So sqrt(4/9) = 2/3.