EXERCISES AND PROBLEMS

DO YOU KNOW THE BASICS?

Number sets a) Which are integers? b) Are there any natural numbers? c) Are there any irrational numbers?

1 Look at the following numbers. // ,, 23 3567 17 18 26 16 18 ;; ;; ;; !

Powers, roots and radicals

9 Calculate the value of these expressions:

25 / 52 b)

/ 43 c) 81 ,

10 Express in exponential form.

2 a) Classify into rational and irrational numbers. b) Put them in order from smallest to biggest. c) Which are real numbers?

;, ;; ;; ;π e 2 3 08 74 3 7 2 1 2 3 !

3 Find an integer, a non-integer rational number and an irrational number between 29 – and 8 –

4 Indicate which of the sets (N, Z, Q or Á) these numbers belong to: π , 7 4 6 13 52 152 2 13 –; ;; ;; ; + !

Intervals and half-lines

5 Represent each of the following intervals and halflines on the real number line:

A = [–2, 4] B = (1, 6) C = [–7, –3)

D = (0, 5] E = (– ∞, 1] F = (–1, +∞)

6 Write the following sets of numbers in interval or half-line form: a) Greater than 2 and less than 7. b) Between –1 and 3, both included. c) Greater than or equal to 5. d) Less than 10.

7 Represent each of the following intervals and halflines as the real number line: a) –3 ≤ x ≤ 2 b) 5 < x c) x ≥ –2 d) –2 ≤ x < 3/2 e) 4 < x < 4.1 f ) –3 ≤ x

8 Express each of the following sets of numbers as an interval or half-line and as an inequality. a) –1 0 3 b) 1 5 c) a) 51/2 b) (–3)2/3 c) d) (a 3)1/4 e) (a 1/2)1/3 f ) (a –1)3/5

11 Write as roots.

12 Express each of these numbers as a power with the base 2 and then find their product: a) 16 / 32 3 –`j b) 4

13 Express the following radicals as powers with fractional exponents and simplify:

14 Simplify.

Approximate numbers. Scientific notation

15 Find the absolute and relative errors in these approximations: a) 3 41 to two decimal places b) 72 to one decimal place c) 17 123 to one hundredth

16 Give a limit of absolute error and of relative error for these approximations of the number of votes in an election: a) 348 thousand votes b) 5 632 000 c) 890 000 d) 78945 e) 9 million f ) 134 thousand a) (1.5 · 107) · (2 · 105) b) (3 · 106) : (2 · 1011) c) (4 · 10–7) : (2 · 10–12) d) · 410 8 a) (3.5 · 107) · (4 · 108) b) (5 · 10–8) · (2.5 · 105) c) (1.2 · 107) : (5 · 10–6) d) (6 · 10–7)2

17 Calculate mentally.

18 Use scientific notations to perform these calculations and give a limit of absolute error.

Logarithms

19 Apply the definition of logarithm and calculate.

25 Express each of the sets of numbers below as a union of intervals, using the symbol ∪:

20 Calculate the base of the following logarithms: a) logb 64 = 3 b) logb 100 = 2 c) logb 243 = 5 d) logb 625 = 4

21 Use a calculator to find: a) log4 23.4 b) log3 543 c) log

20.8

Practice Makes Perfect

22 Place the following numbers on a diagram like this one:

1; , 723 # ; 1 – 2 ;

3.5; 9 11 ; 4 1 ;

6 ; π 4 ; –104

23 a) Which irrational numbers are represented by points A, B, C and D ?

27 Express as powers and calculate x in each case, equalling the exponents of the two sides: b) Represent 8 and 11 .

24 a) Indicate which of the following numbers are included in A = [–3, 7) or in B = (5, +∞): –3; 10; 0,5; 7; – 4; 5 ; , 63 ! ; π; 5 27 ; 48 ; 1 – 2 b) Which of these intervals represents the numbers included in A and in B ?

(–3, 5) [2, 7) [5, 7] (5, 7) c) Express A ∪ B and A ∩ B as intervals and as inequalities.

Take out all possible factors

30 Reduce to a common index and express the results with a single radical.