Rational and Irrational Numbers: math practice, Grade 8 – download the PDF

1 Sort the numbers ★★★

Sort these numbers into rational and irrational: \(5\), \(-\dfrac{2}{3}\), \(\sqrt{49}\), \(\sqrt{11}\), \(0.\overline{3}\), \(2.75\), \(\pi\).

2 Perfect squares and cubes ★★★

Evaluate: a) \(\sqrt{81}\) b) \(\sqrt{144}\) c) \(\sqrt{\dfrac{1}{4}}\) d) \(\sqrt{0.36}\) e) \(\sqrt[3]{27}\) f) \(\sqrt[3]{-64}\) g) \(\sqrt[3]{\dfrac{8}{27}}\)

3 Fractions to decimals ★★★

Write each fraction as a decimal and say whether it terminates or repeats: a) \(\dfrac{3}{8}\) b) \(\dfrac{5}{9}\) c) \(\dfrac{7}{11}\) d) \(\dfrac{1}{6}\)

4 Between two integers ★★★

Between which two consecutive integers does each number lie? a) \(\sqrt{20}\) b) \(\sqrt{50}\) c) \(\sqrt{90}\) d) \(\sqrt{5}\) e) \(\sqrt[3]{30}\) f) \(\sqrt[3]{100}\)

5 True or false? ★★★

True or false? Justify each answer. a) Every integer is a rational number. b) \(\sqrt{25}\) is irrational. c) \(\pi = \dfrac{22}{7}\). d) A terminating decimal is rational. e) A number can be both rational and irrational.

6 Solve the equations ★★★

Solve: a) \(x^2 = 49\) b) \(x^2 = 2.25\) c) \(x^3 = -8\) d) \(x^3 = \dfrac{1}{125}\) e) \(x^2 = 17\) f) \(x^3 = 20\). Leave irrational answers in radical form.

7 Garden, box and tile ★★★

a) A square garden has an area of \(169\) square feet. How long is each side? Give the answer in feet and in meters (1 ft = 0.3048 m). b) A cube-shaped box has a volume of \(216\) cubic inches. Find its edge. c) A square tile has an area of \(50 \text{ cm}^2\). Find its side to the nearest hundredth.

8 Repeating decimals to fractions ★★★

Write each repeating decimal as a fraction in lowest terms: a) \(0.\overline{27}\) b) \(0.\overline{504}\) c) \(1.\overline{4}\) d) \(0.\overline{12}\)

9 A digit in front of the block ★★★

Convert to fractions: a) \(0.2\overline{7}\) b) \(0.8\overline{3}\) c) \(0.1\overline{45}\)

10 Squeeze the root ★★★

Use squares to find the two tenths that surround each number. Then give the best estimate. a) \(\sqrt{13}\) b) \(\sqrt{41}\) c) \(\sqrt{75}\) (to the nearest hundredth)

11 Which is bigger? ★★★

Compare using \(\lt\) or \(\gt\), and justify with squares or cubes when possible. a) \(\sqrt{30}\) and \(5.5\) b) \(\sqrt{12}\) and \(\dfrac{7}{2}\) c) \(\sqrt[3]{9}\) and \(2.1\) d) \(\pi\) and \(\dfrac{22}{7}\)

12 Rational or irrational results ★★★

Decide whether each result is rational or irrational. a) \(\sqrt{9} + \sqrt{16}\) b) \(\sqrt{2} \times \sqrt{8}\) c) \(2\pi\) d) \(\sqrt{0.09}\) e) \(4 + \sqrt{5}\)

13 Name that point ★★★

The points \(A\) to \(E\) are plotted below. Match each one with one of these numbers: \(\sqrt{2}\), \(\sqrt{5}\), \(\sqrt[3]{20}\), \(-\sqrt{3}\), \(\sqrt{12}\).

14 The aquarium ★★★

A cube-shaped aquarium holds \(64\) liters of water. (1 liter = \(1000 \text{ cm}^3\)). Find the length of one edge in centimeters, and then in inches (1 in = 2.54 cm), to the nearest tenth.

15 Pattern or no pattern? ★★★

Decide if each number is rational or irrational, and explain. a) \(0.101001000100001\ldots\) (one more zero each time) b) \(0.\overline{142857}\) c) \(0.12122122212222\ldots\) (one more \(2\) each time) d) \(3.14\)

16 A tricky repeating decimal ★★★

Write \(2.0\overline{36}\) as a fraction in lowest terms and check your answer with a division.

17 Is 0.999… equal to 1? ★★★

Some students believe that \(0.\overline{9}\) is a little smaller than \(1\). Use the conversion method to settle the question. Then explain what this says about \(0.\overline{9}\) being rational.

18 Order the negatives ★★★

Order from least to greatest: \(-\sqrt{7}\), \(-2.6\), \(-\dfrac{5}{2}\), \(-\sqrt[3]{15}\), \(-2.4\).

19 How many integers? ★★★

a) How many integers \(n\) satisfy \(7 \lt \sqrt{n} \lt 8\)? List the smallest and the largest. b) How many integers \(n\) satisfy \(6.2 \lt \sqrt{n} \lt 6.3\)?

20 The hypotenuse ★★★

A ramp makes a right triangle with legs of \(9\) feet and \(5\) feet. a) Write the exact length of the hypotenuse. b) Between which two tenths does it lie? c) Is the length rational or irrational? Explain.

21 Find the mistakes ★★★

A student wrote four statements. Find the error in each and correct it. a) \(\sqrt{50} = 25\) b) \(0.\overline{45} = \dfrac{45}{100}\) c) \(\sqrt{20}\) is between \(5\) and \(6\) d) \(\sqrt[3]{-8}\) does not exist.