Representing real numbers on the number line

First evaluate the square root: \(\sqrt{64}=8\). Therefore, \(\frac{\sqrt{64}-3}{5}=\frac{8-3}{5}=\frac{5}{5}=1\). Hence, the expression represents the point 1 on the number line. Option D results from dividing 3 by 5 without correctly evaluating the numerator; evaluate the square root and subtraction before dividing.

Moving ( \frac{5}{4} ) to the right of (-2) gives ( -2+\frac{5}{4}=-\frac{3}{4} ). Use the given interval to choose the direction.

For a nonnegative fraction, the principal square root can be found by taking the positive square root of the numerator and denominator separately. Since \(49=7^2\) and \(121=11^2\), we have \(\sqrt{49/121}=\sqrt{49}/\sqrt{121}=7/11\). Thus the point is represented by \(7/11\), which is option A.

The positive sign matters because the symbol \(\sqrt{\,}\) denotes the principal, nonnegative square root. The value \(49/11\) forgets to take the square root of the denominator, while \(11/7\) reverses the fraction. Also, \(7/121\) is not the result of taking either required square root. Therefore option A follows.

Since \(\sqrt{3}\approx1.732\), we get \(\sqrt{3}+\frac{1}{4}\approx1.982\). Thus, \(1.99>1.982\) and \(1.99<2\), so 1.99 is correct. The values 1.95 and 1.90 are smaller than the expression, while 2.10 is greater than 2. Exam tip: First find a suitable decimal approximation of the irrational expression, then compare it with the given values.

Since \(\sqrt{28}\approx 5.292\), we get \(A\approx -5.292\). Also, \(B=-5.3\) and \(C=-\frac{16}{3}\approx -5.333\). Among negative numbers, the smaller number lies farther to the left on the number line. Thus, \(-5.333<-5.3<-5.292\), so \(C\) is farthest left. Exam tip: When comparing negative numbers, the one with the greater magnitude is the smaller number.

On a number line, the distance of a point x from zero is |x|. The equation |x| = √17 has two solutions: x = √17 and x = −√17. The additional condition says that the point must lie to the left of zero. Every point left of zero has a negative coordinate, so the required solution is −√17. Therefore option A is correct. Option B has the same distance from zero but lies to the right. Option C has distance 17, not √17, and option D has distance 17 as well, despite being negative. The direction condition is essential for choosing between the two points.

To compare the two numbers, rewrite \(3-\sqrt{2}<\frac{8}{5}\) as \(\sqrt{2}>\frac{7}{5}\). Both sides are positive, and \(2=\frac{50}{25}>\frac{49}{25}=\left(\frac{7}{5}\right)^2\), so \(\sqrt{2}>\frac{7}{5}\). Hence \(3-\sqrt{2}<\frac{8}{5}\). Their approximate values, \(1.586\) and \(1.6\), confirm this; therefore option B is incorrect. Exam tip: check that both sides are non-negative before squaring an inequality.

( \sqrt{2}\approx1.414 ), so ( \frac{a}{10} ) must be between (1.414) and (1.5). (a=14.5) gives (1.45).

Since \(80=16\times5\) and 16 is the largest perfect-square factor of 80, \(\sqrt{80}=\sqrt{16\times5}=4\sqrt{5}\). Option B, \(2\sqrt{20}\), is equivalent but not in simplest form. Also, \(4\sqrt{5}\approx8.94\), so the number lies between 8 and 9 on the number line. Exam tip: when simplifying a square root, extract the largest perfect-square factor.

Since ( -1.42<-1.414<-1.4 ), the order is ( -1.42,-\sqrt{2},-\frac{7}{5} ). For negative decimals the smaller value comes first.

Since \\(5.2<x<5.3\\) and both bounds are positive, squaring preserves the inequality: \\(5.2^2<a<5.3^2\\). Thus, \\(27.04<a<28.09\\). Among the given options, only 28 lies in this interval; 26 and 25 are too small, while 30 is too large. Exam tip: when both sides of an inequality are positive, squaring does not reverse its order.

( \sqrt{7}\approx2.646 ) and ( \sqrt{5}\approx2.236 ), so the difference is about (0.41). The difference of nearby roots is small.

( -\sqrt{3}\approx-1.732 ), and (-1.8) lies between it and (-2). Check left-to-right order in negative intervals.

Since \(\sqrt{49}=7\) and \(\sqrt{16}=4\), we get \(\sqrt{49}+\sqrt{16}=7+4=11\). Therefore, the coordinate of point \(P\) on the number line is \(11\). The value 65 is the product \(7\times 4\), not the required sum. Exam tip: evaluate each square root first and then perform the indicated operation.

The governing concept is simplifying a real-number expression before locating its value on the number line. The principal square root of 4 is √4 = 2, because 2² = 4 and the principal root is non-negative. Therefore, 1/√4 = 1/2. This point is positive and lies halfway between 0 and 1, so option A is correct. Option B is the denominator after simplification, not its reciprocal. Option C would result from incorrectly replacing √4 by 4 and then taking 1/4. Option D is the radicand itself, not the value of the given expression. The exact simplification leaves no need for approximation.

\(C=-\frac{25}{4}=-(25\div 4)=-6.25\), so \(A=C\) and both points represent the same position on the number line. \(B=-\sqrt{39}\) is different because \(\sqrt{39}\neq 6.25\); in fact, \(6.25^2=39.0625\), not \(39\). Exam tip: Convert a fraction to decimal form when comparing it with a terminating decimal.

( \sqrt{90}\approx9.49 ), which is closer to (10) than to (9). Check distance for the nearest integer.

The governing concept is comparison of negative real numbers within an inequality. First, √50 = 5√2 ≈ 7.071, so -√50 ≈ -7.071. The condition therefore requires x to be greater than -7.071 but less than -7; in interval notation, x must belong to approximately (-7.071, -7). The value -7.05 lies inside this narrow interval, so option A is possible. The value -7.20 is less than -7.071 and lies outside on the left. The value -6.90 is greater than -7, and -8.00 is also smaller than the left endpoint. Because negative numbers increase when they move rightward on the number line, careful order checking is essential.

Adding like radicals gives ( \sqrt{13}+\sqrt{13}=2\sqrt{13} ). Do not add the numbers inside the radicals directly.

Given \(\sqrt{x}=3.9\), squaring both sides gives \(x=(3.9)^2=3.9\times3.9=15.21\). Hence, \(x\) represents the point 15.21 on the number line. The values 7.8, 12.21, and 3.09 do not result from correctly squaring 3.9. Exam tip: when squaring a decimal, multiply the numbers as whole numbers first and then place the decimal point according to the total number of decimal places.