Representing real numbers on the number line

The expression is \\(\sqrt{\frac{81}{196}}\\). Since both 81 and 196 are perfect squares, take their positive square roots: \\(\sqrt{81}=9\\) and \\(\sqrt{196}=14\\). Therefore \\(\sqrt{\frac{81}{196}}=\frac{\sqrt{81}}{\sqrt{196}}=\frac{9}{14}\\). The principal square root is nonnegative, so the result is positive. Hence option A represents the correct point on the number line. Options such as \\(\frac{81}{14}\\) or \\(\frac{9}{196}\\) do not result from taking the square root of numerator and denominator.

The fraction is already in lowest terms because 9 and 14 have no common factor greater than 1. Squaring the answer also verifies it: \\(\left(\frac{9}{14}\right)^2=\frac{81}{196}\\). Thus the point corresponding to the given radical is exactly \\(\frac{9}{14}\\), not a negative value or an unreduced expression. The supplied answer A is mathematically consistent and follows from the square-root property for positive numerator and denominator.

Since \(108=36\times3\) and \(36\) is the greatest perfect-square factor of \(108\), \(\sqrt{108}=\sqrt{36\times3}=6\sqrt{3}\). Thus, the point’s coordinate on the number line is \(6\sqrt{3}\). Exam tip: extract the greatest perfect-square factor from under the radical. Option B does not represent the correct simplification, while options C and D give larger values than \(\sqrt{108}\).

\(-0.3125=-\frac{3125}{10000}\). Dividing the numerator and denominator by 625 gives \(-\frac{3125}{10000}=-\frac{5}{16}\), so option A is correct. The closest distractor, \(-\frac{3}{16}\), equals \(-0.1875\), not \(-0.3125\). Exam tip: for a terminating decimal, place it over the corresponding power of 10 and then reduce the fraction completely.

( \sqrt{11}\approx3.316 ) and ( \frac{10}{3}\approx3.333 ), so (3.33) lies between them. Compare very close values accurately.

The direct answer is option A, 1.5. A length on the number line is found by taking the absolute difference of the endpoint coordinates. For AB, use \(AB=|A-B|=|2.4-(-1.8)|=|2.4+1.8|=4.2\). For AC, use \(AC=|A-C|=|2.4-5.1|=|-2.7|=2.7\). The difference between the lengths is \(|4.2-2.7|=1.5\). Option A is therefore correct. Option B, 2.7, is only the length AC, not the difference between AB and AC. Option C, 4.2, is only the length AB, not the requested difference. Option D, 6.9, comes from adding the two lengths, \(4.2+2.7\), whereas the question asks for their difference. Absolute value is important because a length cannot be negative. The points' order does not matter when calculating distance. Memory cue: first calculate both lengths with absolute values, then subtract the two lengths and take the positive result.

The direct answer is option B: 1.55. First evaluate the two square roots: \(\sqrt{2.25}=1.5\), because \(1.5^2=2.25\); and \(\sqrt{2.56}=1.6\), because \(1.6^2=2.56\). Thus the question asks for a value strictly between 1.5 and 1.6. Now check the options: 1.4 is below 1.5, 1.55 is between 1.5 and 1.6, 1.7 is above 1.6, and 1.2 is below 1.5. Therefore option B is correct. Option A is too small, option C is too large, and option D is also too small. The decimal points do not change the method: find the positive square roots and compare ordinary decimal values. Memory cue: square the endpoints mentally; \(1.5^2=2.25\) and \(1.6^2=2.56\), so any answer between 1.5 and 1.6 works, and only 1.55 is offered.

( \frac{13}{3}\approx4.333 ), ( \sqrt{19}\approx4.359 ), and (4.34) lies between them. The correct order is ( \frac{13}{3},4.34,\sqrt{19} ).

Given \(\sqrt{x}=4.7\), squaring both sides gives \(x=(4.7)^2=22.09\). Therefore, \(x\) represents 22.09 on the number line. The nearby option 21.09 is incorrect because \(4.7\times4.7=22.09\), not 21.09. Exam tip: when squaring a decimal, multiply it carefully and place the decimal point according to the total number of decimal places in the factors.

( -\sqrt{13}\approx-3.606 ), so (-3.65) is less than it and greater than (-3.7). Be careful with direction for negative values.

( \sqrt{12}=2\sqrt{3} ) and ( \sqrt{27}=3\sqrt{3} ), so the sum is (5\sqrt{3}). Simplify the radicals first.

Since \(\sqrt{7}\approx 2.646\), we get \(u=-2-2.646\approx -4.646\). Therefore, \(u\) lies between \(-5\) and \(-4\). Note that subtracting a positive number from a negative number makes the result more negative, so it cannot lie between \(-4\) and \(-3\). In an exam, estimate the square root first and then locate the resulting number between consecutive integers.

( \sqrt{6}\approx2.449 ) and ( \frac{1}{3}\approx0.333 ), so the sum is about (2.782). For mixed values, estimate first.