Representing real numbers on the number line

(7.5^2=56.25) and (7.6^2=57.76), so ( \sqrt{57} ) lies between them. Check squares for decimal bounds.

Evaluate the square root first: \(\sqrt{121}=11\). Thus, \(\frac{\sqrt{121}-4}{7}=\frac{11-4}{7}=\frac{7}{7}=1\). Therefore, the corresponding coordinate on the number line is \(1\). Option D results from dividing only 4 by 7 and ignores the subtraction involving \(\sqrt{121}\). Exam tip: follow the order of operations—evaluate the square root, simplify the numerator, and then divide.

Moving ( \frac{7}{5} ) to the right of (-3) gives ( -3+\frac{7}{5}=-\frac{8}{5} ). Use the given interval to choose direction.

Since \(169=13^2\) and \(225=15^2\), \(\sqrt{\frac{169}{225}}=\frac{\sqrt{169}}{\sqrt{225}}=\frac{13}{15}\). The principal square root is non-negative, so the positive values 13 and 15 are used. Option B is the original fraction, not its square root. Exam tip: For a fraction whose numerator and denominator are perfect squares, take the positive square root of each separately.

( \sqrt{8}+\frac{1}{10}\approx2.928 ). Therefore (2.90) does not satisfy it and no given value satisfies the condition.

Since
C=-\frac{43}{4}=-10.75
, points B and C coincide. Also,
\sqrt{116}\approx10.7703
, so
A=-\sqrt{116}\approx-10.7703
. Among negative numbers, the smaller number lies farther left on the number line; therefore, point A is farthest left. Exam tip: When comparing negative numbers, the one with the greater absolute value is the smaller number.

The governing concept is absolute distance on a number line. A point at distance \(d\) from zero can be either \(d\) or \(-d\), because both have absolute value \(d\). Here the distance is \(\sqrt{26}\), so the two possible points are \(\sqrt{26}\) and \(-\sqrt{26}\). The condition that the point lies to the right of zero selects the positive value, \(\sqrt{26}\). Therefore option B is correct. Option A has the required distance but lies left of zero; options C and D have distance 26, not \(\sqrt{26}\).

(4-\sqrt{6}\approx1.551) and ( \frac{31}{20}=1.55 ), so the first value is slightly greater. Estimate close values accurately.

( \sqrt{2}\approx1.414 ), so ( \frac{a}{100} ) must be between (1.414) and (1.42). (a=141) gives (1.41), which is not correct.

Since \(180=36\times5\) and \(36\) is the largest perfect-square factor, \(\sqrt{180}=\sqrt{36\times5}=6\sqrt{5}\). Option A, \(3\sqrt{20}\), is equivalent but not fully simplified because \(20\) still contains the square factor \(4\). For such questions, take the largest perfect-square factor outside the radical.

( -\frac{53}{20}=-2.65 ), ( -\sqrt{7}\approx-2.646 ), and (-2.64). For negative values, the smallest number comes first.

Since \(x=\sqrt{a}\) and both bounds are positive, squaring preserves the inequality: \(11.3^2<a<11.4^2\), so \(127.69<a<129.96\). Among the given choices, only \(128\) lies in this interval; \(127\) is too small and \(131\) is too large. Exam tip: for a square-root inequality with positive bounds, square the boundary values to find the interval for the radicand.

( \sqrt{11}\approx3.317 ) and ( \sqrt{10}\approx3.162 ), so the difference is about (0.155). The difference of nearby roots is small.

( -\sqrt{15}\approx-3.873 ), so (-3.95) lies between (-4) and (-3.873). Read negative intervals in order.

Since \(\sqrt{196}=14\) and \(\sqrt{81}=9\), we get \(\sqrt{196}-\sqrt{81}=14-9=5\). Therefore, the coordinate of P is 5. Option D gives only the first square root and ignores the subtraction. Exam tip: evaluate the square roots first, then perform the indicated operation.

( \sqrt{25}=5 ), so ( \frac{1}{\sqrt{25}}=\frac{1}{5} ). Simplify the root in the denominator first.

\(-\frac{57}{8}=-7.125\), so \(A\) and \(C\) represent the same point on the number line. However, \(-\sqrt{51}\) is different because \(\left(\frac{57}{8}\right)^2=\frac{3249}{64}\neq 51\). Exam tip: To compare a fraction and a decimal, convert the fraction to a decimal or compare their squares when both numbers are negative radicals or rational values.

( \sqrt{132}\approx11.49 ), so it is slightly closer to (11). Check distance for the nearest integer.

The governing concept is evaluating the position of a negative square root before testing an inequality. Since \(9^2=81<82<100=10^2\), we have \(9<\sqrt{82}<10\), and more closely \(\sqrt{82}\approx9.055\). Therefore \(-\sqrt{82}\approx-9.055\), so x must be greater than -9.055 and less than -9. The value -9.03 satisfies both conditions. The value -9.20 is too small, -8.90 is greater than -9, and -10.00 is far below the allowed interval. Hence option A is correct.

Adding like radicals gives ( \sqrt{19}+\sqrt{19}+\sqrt{19}=3\sqrt{19} ). Do not add the numbers inside radicals directly.