Representing real numbers on the number line

( \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.

The correct answer is option C: \(\frac{1}{5}\). Start with the denominator: \(\sqrt{25}=5\), because \(5\times5=25\) and the principal square root is positive. Therefore \(\frac{1}{\sqrt{25}}=\frac{1}{5}\). On the number line this is the point one-fifth of a unit from zero, approximately 0.2. Option A, \(\frac{1}{25}\), would result from incorrectly treating \(\sqrt{25}\) as 25; it is approximately 0.04 and is not the given value. Option B is 5, but the expression is the reciprocal of 5, not 5 itself. Option C is correct after simplifying the root. Option D is 25 and makes the same root mistake as option A. First simplify the square root, then take the reciprocal. Memory cue: \(\sqrt{n^2}=n\) for positive n, so \(1/\sqrt{25}=1/5\), not \(1/25\).

\(-\frac{57}{8}=-7.125\), 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.

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

Given \(\sqrt{x}=6.8\), squaring both sides gives \(x=(6.8)^2=46.24\). Therefore, \(x\) represents the point 46.24 on the number line. The closest distractor, 45.24, is not the square of 6.8. Exam tip: For a decimal square, use \((a+b)^2=a^2+2ab+b^2\); here, \((6+0.8)^2=36+9.6+0.64=46.24\).

\(\frac{17}{5}=3.4\) and \(\sqrt{12}\approx3.464\). Thus, \(3.4<3.45<3.464\), so 3.45 is the correct value. Options 3.47 and 3.50 are greater than \(\sqrt{12}\), while 3.30 is less than \(\frac{17}{5}\). Exam tip: Convert the fraction to a decimal and estimate the square root before comparing the values.

( -\sqrt{18}-2\approx-6.243 ), so it lies between (-7) and (-6). Estimate negative sums carefully.

( \sqrt{192}=8\sqrt{3} ) and ( \sqrt{75}=5\sqrt{3} ), so the difference is (3\sqrt{3}). Simplify the radicals first.

( \sqrt{126}\approx11.22 ) and ( \sqrt{80}\approx8.94 ), so the difference is about (2.28). Estimate both square roots first.

( |x-1.5|=4.25 ) means the distance of (x) from (1.5) is (4.25). Moving both directions gives ( -2.75 ) and (5.75).

( \frac{41}{10}=4.1 ), ( \sqrt{17}\approx4.123 ), and (4.13). Convert close values to decimals for comparison.

( -\sqrt{93}\approx-9.644 ) and (-9.62) is greater. The greater number lies farther right on the number line.

Since \(2.8^2=2.8\times2.8=7.84\), we have \(\sqrt{7.84}=2.8\). The nearby choices are incorrect because \(2.7^2=7.29\) and \(2.9^2=8.41\). In an exam, squaring the options is a quick way to verify a decimal square root.

The distance is \( \left|\frac{9}{10}-\left(-\frac{17}{5}\right)\right|=\frac{43}{10} \). Use absolute value while finding distance.

( \sqrt{48}\approx6.928 ) and ( \sqrt{49}=7 ), so (6.95) lies between them. Use accurate estimation for close roots.

( \sqrt{39}\approx6.245 ), so (6-\sqrt{39}\approx-0.245). Always check the sign in root subtraction.

Since \(2.56=(1.6)^2\), we have \(\sqrt{2.56}=1.6\). The negative sign outside the radical changes the entire value to negative, so \(-\sqrt{2.56}=-1.6\). Option B omits the negative sign, while C and D result from misreading the decimal or the radicand as the square root. Exam tip: find the principal square root first, then apply any sign outside the radical.

Since \(169=13^2\), we have \(\sqrt{169}=13\). Therefore, \(x=\frac{13}{6}\), so this is the point represented on the number line. Option B incorrectly leaves 169 in the numerator, while option C reverses the fraction. Exam tip: simplify the square root first and then perform the remaining operation.