Representing real numbers on the number line

Correct answer: B. ( -4.90 )

( -\sqrt{23}\approx-4.796 ), so (-4.90) lies between (-5) and (-4.796). Read negative intervals in order.

Correct answer: C. ( \frac{1}{6} )

The correct answer is option C: \(\frac{1}{6}\). Simplify the denominator first: \(\sqrt{36}=6\), since \(6^2=36\). Hence \(\frac{1}{\sqrt{36}}=\frac{1}{6}\). This represents a positive point approximately 0.1667 on the number line. Option A, \(\frac{1}{36}\), incorrectly treats the square root as 36; it is much smaller than the required value. Option B gives 6, but the expression is the reciprocal of 6, not 6. Option C correctly applies the reciprocal after simplifying the root. Option D gives 36 and makes the same basic root error as option A. A useful order is: evaluate the root, then keep the numerator 1 over that result. Memory cue: \(\sqrt{36}=6\), so the reciprocal is \(1/6\), never \(1/36\).

Correct answer: C. A and C

A and C are equal because −67 ÷ 8 = −8.375. Also, √70 is approximately 8.367, so B is approximately −8.367, not −8.375. Therefore, A and C represent the same value.

Correct answer: B. (12)

( \sqrt{156}\approx12.49 ), so it is slightly closer to (12). Check distance for the nearest integer.

Correct answer: A. ( -11.03 )

( -\sqrt{122}\approx-11.045 ), so (x) must lie between (-11.045) and (-11). (-11.03) is correct.

Correct answer: B. (4\sqrt{29})

Adding like radicals gives ( \sqrt{29}+\sqrt{29}+\sqrt{29}+\sqrt{29}=4\sqrt{29} ). Do not add the numbers inside radicals directly.

Correct answer: C. 62.41

Given √x = 7.9, squaring both sides gives x = (7.9)² = 62.41. Therefore, x is represented by the point 62.41 on the number line. The nearby option 61.41 is incorrect because it does not equal the square of 7.9. Exam tip: calculate a decimal square as 7.9 × 7.9 or as (7 + 0.9)².

Correct answer: B. Between ( -8 ) and ( -7 )

( -\sqrt{27}\approx-5.196 ), so ( -\sqrt{27}-3\approx-8.196 ). Therefore it lies between (-9) and (-8).

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

( \sqrt{300}=10\sqrt{3} ) and ( \sqrt{147}=7\sqrt{3} ), so the difference is (3\sqrt{3}). Simplify the radicals first.

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

The midpoint is ( \frac{\sqrt{2}+\sqrt{8}}{2}=\frac{\sqrt{2}+2\sqrt{2}}{2}=\frac{3\sqrt{2}}{2} ). Take the average of the two values for the midpoint.