Representing real numbers on the number line

( \sqrt{18}\approx4.243 ) and ( \sqrt{17}\approx4.123 ), so the difference is about (0.12). The difference of nearby roots is small.

( -\sqrt{23}\approx-4.796 ), so (-4.90) lies between (-5) and (-4.796). Read negative intervals in order.

The governing concept is evaluating exact square roots and then performing the indicated subtraction. Since 256 = 16², √256 = 16. Since 121 = 11², √121 = 11. Therefore the coordinate of P is √256 − √121 = 16 − 11 = 5, so option A is correct. No decimal approximation is needed because both radicands are perfect squares. Option B comes from adding the two roots instead of subtracting them. Option C has no valid basis in the calculation, while option D is only √256 and ignores the second term. The coordinate is therefore an exact real number, and its positive value places P five units to the right of zero on the number line.

( \sqrt{36}=6 ), so ( \frac{1}{\sqrt{36}}=\frac{1}{6} ). Simplify the root in the denominator first.

c(-\frac{67}{8}=-8.375c), so A and C represent the same value. In contrast, c(-\sqrt{70}c) is approximately c(-8.367c), so it is not equal to c(-8.375c). Exam tip: convert the fraction to a decimal, or square the decimal to compare it with the radicand.

( \sqrt{156}\approx12.49 ), so it is slightly closer to (12). Check distance for the nearest integer.

( -\sqrt{122}\approx-11.045 ), so (x) must lie between (-11.045) and (-11). (-11.03) is correct.

Adding like radicals gives ( \sqrt{29}+\sqrt{29}+\sqrt{29}+\sqrt{29}=4\sqrt{29} ). Do not add the numbers inside radicals directly.

Given √x = 7.9, squaring both sides gives x = (7.9)² = 62.41. Therefore, x is represented by the point 62.41 on the number line. The nearby option 61.41 is incorrect because it does not equal the square of 7.9. Exam tip: calculate a decimal square as 7.9 × 7.9 or as (7 + 0.9)².

( -\sqrt{27}\approx-5.196 ), so ( -\sqrt{27}-3\approx-8.196 ). Therefore it lies between (-9) and (-8).

( \sqrt{300}=10\sqrt{3} ) and ( \sqrt{147}=7\sqrt{3} ), so the difference is (3\sqrt{3}). Simplify the radicals first.

The midpoint is ( \frac{\sqrt{2}+\sqrt{8}}{2}=\frac{\sqrt{2}+2\sqrt{2}}{2}=\frac{3\sqrt{2}}{2} ). Take the average of the two values for the midpoint.