At a space observatory, Emek and Luna found the viewing dome locked just before a distant planet was due to appear. The control screen displayed four coordinates: 0.75, sqrt(81), sqrt(18), and 7/3. A message instructed them to enter the rational coordinates first, from least to greatest, followed by the irrational coordinate. Luna recognized 0.75 as rational because it equals 3/4, and 7/3 was already written as a ratio of integers. Emek simplified sqrt(81) to 9, making it rational too. However, sqrt(18) was irrational because 18 is not a perfect square. To order the rational values, they compared 0.75, 7/3, and 9 and entered them in that sequence. Then they added sqrt(18) as the final coordinate. Emek checked every classification before Luna activated the controls. The sequence was accepted, the dome slowly opened, and they began observing the distant planet.

Challenges

Start

Challenge 1 of 5

Challenges

Start

Challenge 1 of 4

Hint

The classification depends on the value of sqrt(18), not on whether 18 itself is an integer.

A student claims that sqrt(18) should be placed in the rational group because 18 is an integer. Which statement correctly identifies the student's mistake?

Every square root is irrational, even when the number inside is a perfect square.A square root is rational only when the number inside it is even.The student should classify sqrt(18) as rational because 18 can be written as 18/1.The fact that 18 is an integer does not make sqrt(18) rational; since 18 is not a perfect square, sqrt(18) is irrational.

Correct! Since 18 is not a perfect square, sqrt(18) is irrational.

Not quite. Consider the value of the square root itself and whether it can be written as a ratio of integers.

🎉 Congrats! You got this question!

Hint

The story identifies three original coordinates as rational. Classify each of the three new coordinates and update the count.

The control screen adds three more coordinates: sqrt(64), sqrt(20), and 2.5. Including the original coordinates 0.75, sqrt(81), sqrt(18), and 7/3, how many of the seven coordinates are rational?

Correct! Five of the seven coordinates are rational.

Check whether each new coordinate can be written as a ratio of integers, then combine that result with the original classifications.

🎉 Congrats! You got this question!

Hint

Square the two whole numbers and compare their squares with 18.

Luna estimates that sqrt(18) lies between 4 and 5. Which reasoning correctly verifies her estimate?

4 + 5 = 9, and 9 is less than 18.4^2 = 16 and 5^2 = 25, and 18 lies between 16 and 25.18 / 4 is greater than 4, so sqrt(18) must equal 5.18 is even, so its square root must lie between two even integers.

Correct! Since 16 < 18 < 25, sqrt(18) lies between 4 and 5.

Not quite. Compare 18 with the perfect squares of 4 and 5.

🎉🎉🎉🎉🎉🎉🎉🎉

🎉 Congrats! You finished this challenge set!

Gözlemevi Kubbesinin Kilidini Açmak