Crossing a remote mountain pass, Emek and Tara reached a series of locked checkpoints leading to their expedition camp. Each marker displayed an expression that had to be classified correctly. The first showed sqrt(144) - 17. Tara simplified sqrt(144) to 12, giving -5, an integer and therefore also a rational number. The second displayed 21/6, which Emek reduced to 7/2 and classified as rational but not an integer. At the third checkpoint, sqrt(45) appeared. Tara rewrote it as 3sqrt(5) and explained that it remained irrational because sqrt(5) is irrational. The final marker showed sqrt(81)/3, which simplified to 3, a natural number, integer, rational number, and real number. By recognizing that some number systems are contained within others, Emek and Tara entered every classification correctly. The final gate opened, revealing the trail to the expedition camp.

Challenges

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Challenge 1 of 5

Challenges

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Challenge 1 of 4
Hint
A rational number can be written as a ratio of integers. Check whether 7/2 is also a whole integer.
At the second checkpoint, Emek simplifies 21/6 to 7/2. Which classification is most specific for 7/2?
Natural numberIntegerRational number but not an integerIrrational number
Correct! 7/2 is a ratio of integers, so it is rational, but it is not an integer.
Not quite. Simplify 7/2 and decide whether it is an integer or a non-integer rational number.
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Hint
Simplify each square root first, then subtract the first value from the final value.
The first checkpoint displays sqrt(144) - 17, and the final checkpoint displays sqrt(81)/3. After simplifying both expressions, what is the difference between the final checkpoint value and the first checkpoint value?
Correct! The values are -5 and 3, and 3 - (-5) = 8.
Try again. Simplify both expressions before finding their difference.
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Hint
A square root is rational only when the number under the radical can simplify completely to a rational value.
Emek and Tara compare the third checkpoint, sqrt(45), with the final checkpoint, sqrt(81)/3. Which statement correctly describes the two simplified values?
Both values are irrational because both contain square roots.sqrt(45) simplifies to 3sqrt(5), which is irrational, while sqrt(81)/3 simplifies to 3, which is rational.sqrt(45) is rational because 45 is an integer, while sqrt(81)/3 is irrational.Both values are integers because 45 and 81 are integers.
Correct! sqrt(45) = 3sqrt(5) remains irrational, while sqrt(81)/3 = 3 is rational.
Not quite. Simplify each square root completely before classifying the resulting number.
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Gizli Geçidin Kontrol Noktaları
TR
ARC_G12_NUM_02_STORY_01
Gizli Geçidin Kontrol Noktaları
The Checkpoints of the Hidden Pass
EN
ARC_G12_NUM_02_STORY_01
The Checkpoints of the Hidden Pass