Square root spiral

Correct answer: B. It is between \(5\) and \(6\)

Explanation: Since \(5^2=25\) and \(6^2=36\), and \(25<32<36\), we get \(5<\sqrt{32}<6\). Therefore, the spiral distance \(\sqrt{32}\) lies between \(5\) and \(6\). The option between \(4\) and \(5\) is incorrect because \(\sqrt{32}>\sqrt{25}=5\). Exam tip: estimate a square root by locating the nearest perfect squares on either side of the number.

Correct answer: D. ( \sqrt{17} )

Explanation: ( \sqrt{17} ) is close to (4) because (17) is only (1) more than (16). Look for the number closest to a perfect square.

Correct answer: A. (n,n+1,n+2)

Explanation: Consecutive radii are ( \sqrt{n} ), ( \sqrt{n+1} ), and ( \sqrt{n+2} ). Their squares are (n,n+1,n+2).

Correct answer: B. \(6\sqrt{3}\)

Explanation: Since \(108=36\times3\), and \(36\) is the largest perfect-square factor of 108, \(\sqrt{108}=\sqrt{36}\times\sqrt{3}=6\sqrt{3}\). \(9\sqrt{2}\) is not correct because its square is \(81\times2=162\), not 108. Exam tip: first identify the largest perfect-square factor when simplifying a surd.

Correct answer: B. (11)

Explanation: The direct answer is B: the distance is 11 units. The statement says that the spiral has been constructed up to sqrt{120}, and then sqrt{121} is constructed. Evaluate the new distance: 121=11f11, so sqrt{121}=11. Option A, 10, is not the new distance; 10 is sqrt{100}, an earlier perfect-square value. Option B, 11, is correct because its square is 121. Option C, 12, would have square 144, so it does not represent sqrt{121}. Option D, 121, is the number under the root, not the value of the square root. The key distinction is between the radicand, which is the number inside the root, and the square-root value, which is the distance represented. Thus the construction reaches a point 11 units from the starting point. Memory cue: ask, “What number multiplied by itself gives the number inside the root?” Since 11f11=121, the answer is 11.