Inside an abandoned underground station, Emek and Tara found the navigation system blocking the passage ahead. Its control panel displayed four expressions, and only the rational values could activate the first set of switches. Tara simplified sqrt(50) to 5sqrt(2), which remained irrational. Emek examined sqrt(18) - 3sqrt(2) and rewrote sqrt(18) as 3sqrt(2), making the entire expression equal to 0, a rational integer. Another display showed 4 + sqrt(7), which was irrational because adding a rational number to an irrational number remains irrational. The final expression was sqrt(121)/4, which simplified to 11/4, a rational number but not an integer. They also compared the two rational results and found that 11/4 was greater than 0. Using these classifications and their order, Emek and Tara entered the required code. The navigation system restarted, and the underground passage opened for their expedition.

Challenges

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Challenge 1 of 5

Challenges

Start
Challenge 1 of 4
Hint
Simplify each radical expression and identify which results can be written as ratios of integers.
The navigation system activates switches only for rational values. Which pair of displayed expressions should activate switches?
sqrt(50) and 4 + sqrt(7)sqrt(50) and sqrt(121)/4sqrt(18) - 3sqrt(2) and 4 + sqrt(7)sqrt(18) - 3sqrt(2) and sqrt(121)/4
Correct! The two expressions simplify to 0 and 11/4, both of which are rational.
Not quite. Check whether each radical disappears completely after simplification.
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Hint
There are two rational expressions and two irrational expressions. Find the points for each group and add them.
Emek and Tara found exactly two rational expressions among the four displays. If the system assigns 6 code points for each rational expression and 2 code points for each irrational expression, how many code points are assigned to all four displays?
Correct! The rational expressions contribute 12 points and the irrational expressions contribute 4 points, for 16 total.
Try again. Use the number of rational and irrational expressions separately before adding their code points.
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Hint
Rewrite sqrt(32) by factoring 32 as 16 times 2, then combine like radical terms.
A new display appears with the expression sqrt(32) - 4sqrt(2). Based on the reasoning used for sqrt(18) - 3sqrt(2), how should Emek and Tara classify the new expression?
It equals 0, so it is a rational integer.It equals 4sqrt(2), so it is irrational.It equals sqrt(2), so it is irrational.It equals 4, so it is a rational integer.
Correct! sqrt(32) = 4sqrt(2), so the two terms cancel and the result is 0.
Not quite. Simplify sqrt(32) first, then subtract the like radical terms.
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Yer Altındaki Sayı Kodu
TR
ARC_G12_NUM_02_STORY_02
Yer Altındaki Sayı Kodu
The Underground Number Code
EN
ARC_G12_NUM_02_STORY_02
The Underground Number Code