Square root spiral

Since \(45=9\times5\) and \(9\) is a perfect square, \(\sqrt{45}=\sqrt{9\times5}=\sqrt9\times\sqrt5=3\sqrt5\). Therefore, option C is correct. \(5\sqrt3\) is not correct because its square is \(75\), not \(45\). Exam tip: simplify a surd by taking out its greatest perfect-square factor.

( \sqrt{99} ) is slightly less than (10) because (99) is close to (100). Compare with the perfect square (100).

The quantities \\(OP_a\\) and \\(OP_b\\) have lengths \\(\\sqrt{a}\\) and \\(\\sqrt{b}\\). Squaring them removes the square roots, giving \\(OP_a^2=a\\) and \\(OP_b^2=b\\). Therefore the requested difference is simply the difference between the indexed numbers. Since \\(b=a+5\\), the difference is 5, so option C is correct.

The calculation is \\(OP_b^2-OP_a^2=b-a=(a+5)-a=5\\). No square root remains after squaring the lengths. Option A incorrectly takes the square root of the difference, and option B doubles it. Option D gives the value of b rather than the difference. The result is independent of the particular value of a, provided the stated lengths are defined.

Integer distances occur at perfect squares, and the next perfect square after (28) is (36). So the next integer distance is at ( \sqrt{36}=6 ).

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.

In the right triangle, \( (\sqrt{7})^2+1^2=8 \). This is a direct use of Pythagoras theorem.

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

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

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.

( \sqrt{121}=11 ), so the distance will be (11) units. A perfect square gives an integer square root.