Representing real numbers on the number line

Correct answer: A. ( \sqrt{21}<4.59 )

The correct answer is option A: \(\sqrt{21}<4.59\). Both numbers are positive, so compare their squares. We have \((\sqrt{21})^2=21\), and \(4.59^2=21.0681\). Since \(21<21.0681\), the original positive numbers satisfy \(\sqrt{21}<4.59\). More precisely, \(\sqrt{21}\approx4.582575\), which is just less than 4.59. Option A is therefore correct. Option B reverses the inequality. Option C is wrong because 4.59 is not exactly \(\sqrt{21}\); its square is 21.0681 rather than 21. Option D claims 4.59<4, which is false because 4.59 is greater than 4. On a number line, the square-root point is slightly to the left of 4.59. Memory cue: when comparing positive approximations, squaring both sides is safe and often avoids rounding mistakes.

Correct answer: C. ( -4-\sqrt{23} )

Moving left means subtracting the distance, so the coordinate is ( -4-\sqrt{23} ). Choose the sign by direction.

Correct answer: A. ( \frac{3}{5} )

The midpoint is ( \frac{\frac{7}{15}+\frac{11}{15}}{2}=\frac{3}{5} ). Use the average of the two values for the midpoint.

Correct answer: A. (2\sqrt{3})

( \sqrt{108}=6\sqrt{3} ) and ( \sqrt{48}=4\sqrt{3} ), so the difference is (2\sqrt{3}). Subtract only like radicals.

Correct answer: B. Irrational number

This decimal is non-terminating and non-repeating, so it is irrational. Check whether the decimal pattern truly repeats or not.

Correct answer: C. 71

Because both endpoints are positive, squaring the inequality gives 8.4² < n < 8.5², or 70.56 < n < 72.25. Of the given choices, only 71 lies in this interval, so C is correct.

Correct answer: C. ( -4 ) and ( -3 )

( -\frac{43}{11}\approx-3.909 ), so it lies between (-4) and (-3). Convert negative fractions to decimals.

Correct answer: A. \(-1.125\)

The coordinate of the midpoint is the average of the two coordinates: \(\frac{-4.125+1.875}{2}=\frac{-2.25}{2}=-1.125\). Therefore, option A is correct. Option C results from ignoring the negative sign of A. In an exam, add the coordinates algebraically and divide the result by 2.

Correct answer: A. \(0.15\)

\(0.0225=225/10000\), so \(\sqrt{0.0225}=15/100=0.15\), since \((0.15)^2=0.0225\). In option C, the decimal point is shifted one place too far to the left, making its square much smaller. Exam tip: For a decimal square root, convert the number into a fraction and take the square root of the numerator and denominator separately when both are perfect squares.

Correct answer: B. 13 and 14

Since \(11^2=121<131<144=12^2\), we have \(11<\sqrt{131}<12\). Adding 2 to all parts gives \(13<\sqrt{131}+2<14\), so the expression lies between 13 and 14. Option A is incorrect because it corresponds to the bounds of \(\sqrt{131}\), not of the complete expression. Exam tip: compare the number under the square root with the nearest consecutive perfect squares.

Correct answer: B. Between −6 and −5

Since \(9^2=81<99<100=10^2\), we have \(9<\sqrt{99}<10\). Therefore, \(-10< -\sqrt{99}< -9\), and adding 4 gives \(-6<p<-5\). Hence, \(p\) lies between −6 and −5. Exam tip: when multiplying an inequality by \(-1\), reverse its direction; this prevents choosing the nearby interval −5 to −4.

Correct answer: A. ( \frac{23}{36} )

Moving left gives ( \frac{11}{12}-\frac{5}{18}=\frac{23}{36} ). Subtract the distance according to direction.

Correct answer: C. \(c\)

For comparison, \(\sqrt{26}\approx 5.099\), while \(\frac{51}{10}=5.1\) and \(c=5.11\). Thus, \(5.11>5.1>5.099\), so \(c\) lies farthest to the right on the number line. The values of \(a\) and \(b\) are close, but they are not equal. Exam tip: Convert the numbers to comparable decimal forms before ordering them.

Correct answer: B. (9.2<\sqrt{86}<9.3)

(9.2^2=84.64) and (9.3^2=86.49), so ( \sqrt{86} ) lies between them. Check squares for decimal bounds.

Correct answer: A. \(1\)

First evaluate the square root: \(\sqrt{225}=15\). Therefore, \(\frac{\sqrt{225}-6}{9}=\frac{15-6}{9}=\frac{9}{9}=1\). Hence, the point is \(1\). Exam tip: follow the order of operations carefully; \(\frac{9}{15}\) is incorrect because it reverses the intended numerator and denominator.

Correct answer: A. ( -\frac{17}{6} )

Moving ( \frac{13}{6} ) to the right of (-5) gives ( -5+\frac{13}{6}=-\frac{17}{6} ). Use the given interval to choose direction.

Correct answer: A. \(\frac{14}{17}\)

Since \(196=14^2\) and \(289=17^2\), \(\sqrt{\frac{196}{289}}=\frac{\sqrt{196}}{\sqrt{289}}=\frac{14}{17}\). The square-root symbol denotes the principal, non-negative square root, so \(-\frac{14}{17}\) is not the value. Exam tip: when the numerator and denominator are perfect squares, take their positive square roots separately.

Correct answer: A. (3.95)

( \sqrt{15}+\frac{1}{8}\approx3.998 ), so (3.95) is not greater than it. In this case no listed value satisfies the condition.

Correct answer: A. Point A

Since \(C=-\frac{68}{5}=-13.6\), points \(B\) and \(C\) coincide. Also, \(13.6^2=184.96<185\), so \(\sqrt{185}>13.6\), which gives \(-\sqrt{185}<-13.6\). On a number line, the smaller number lies farther left; hence point \(A\) is farthest left. Exam tip: For negative numbers, the one with the greater absolute value is the smaller number.

Correct answer: A. (5-\sqrt{11}<\frac{17}{10})

To compare the two numbers, estimate the square root. Since \\(\\sqrt{11}\\) is about \\(3.316\\), the first expression is approximately \\(5-3.316=1.684\\). The second expression is \\(\\frac{17}{10}=1.7\\). These estimates are close, so careful calculation is useful, but they clearly show which value is smaller.

More precisely, \\(5-\\sqrt{11}\\approx1.6834\\), while \\(\\frac{17}{10}=1.7\\). Therefore \\(5-\\sqrt{11}<\\frac{17}{10}\\). Both values are positive and near 1.7, so the statement that both are less than \\(-1\\) is impossible. Hence option A is correct. A decimal approximation is sufficient here because the two values are not equal.

Correct answer: B. (173)

( \sqrt{3}\approx1.732 ), so ( \frac{a}{100} ) must be between (1.732) and (1.74). (a=173) gives (1.73), which is slightly smaller, so check the bound carefully.

Correct answer: B. \(7\sqrt{5}\)

Since \(245=49\times5\) and 49 is the largest perfect-square factor, \(\sqrt{245}=\sqrt{49\times5}=7\sqrt{5}\). Option A would represent \(\sqrt{175}\), not \(\sqrt{245}\), and \(\sqrt{49}+\sqrt{5}\) is not equal to \(\sqrt{49+5}\). Exam tip: factor the number using its largest perfect-square factor before simplifying a square root.

Correct answer: A. ( -3.61,-\frac{18}{5},-\sqrt{13} )

By checking carefully, ( -3.61<-\sqrt{13}\approx-3.606<-\frac{18}{5}=-3.6 ). For negative values, the smaller number comes first.

Correct answer: B. \(159\)

Since \(x\) is positive, squaring the inequality gives \(12.6^2<a<12.7^2\). Therefore, \(158.76<a<161.29\). Among the given options, only \(159\) lies in this interval. The value \(158\) is too small, while \(162\) and \(163\) are too large. Exam tip: For \(x=\sqrt a\), square the bounds on \(x\) to find the required interval for \(a\).

Correct answer: B. (0.12)

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