Outside the high school program / Square roots 63 exercises (including 62 corrected)

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4cm2 1. Introduction E.5688 Construct a square whose area is equal to the sum of the areas of the two squares shown opposite. Leave all traces of your research visible. Even if the work is not finished, it will be taken into ac-count in the notation . E.742 1 Which of the expressions below do or do not define a number. Justify your answer. a 1 3 2 − 9 b 2 c − 1 d 0 e ( − 1) 2 f 3 2 2 A student states that : 20 =10 Using only the definition of the square root, justify that his assertion is false. E.238 Give the inverse of each of the following numbers in the most suitable form : a − 0.8 b − 2 3 c 2 d 3 2 e − 0.8 2 f 2 × 0.15 g ( − 0.8) 2 E.272 Simplify the writing of the following ex-pressions : a (2 + 3) 2 b 2 + 12 2 − 8 3 c ( ı − 3) 2 d 81 e ( − 2) 2 f ( 3 − 2) 2 E.237 1 Write A as an irreducible fraction. A = 2 + 1 1 + 1 1 + 1 4 2 a Show that A is an approximate value of 7 to within 10 − 1 . is b A an approximate value of 7 to the nearest 10 − 2 ? E.769 Find counterexamples to the following equalities: a a + b = a + b b a − b = a − b 2. Multiplicative relationships E.749 The definition of the number a is the only positive number such that its square is a . Thus, by defi-nition, the square root of a number verifies : a 2 = a . 1 From the equality 17 2 =289 . Complete the following sen-tences : a The number whose square is 289 is . . . . . . Thus, we have 289 = : : : : : : b The number whose square is 28 900 is . . . . . . Thus, we have 28 900 = : : : : : : c The number whose square is 2.89 is . . . . . . Thus, we have 2.89 = : : : : : : 2 a What is the positive number whose square is 5 2 ? Deduce the value of 5 2 ? b Give the decimal form of the following numbers : A = 12 2 ; B = 1.25 2 ; C = ( − 5) 2 3 a Square the following two numbers : 6 ; 2 × 3 . What can you say about these two numbers? b For each of the pairs, do the same to show the equality of its two numbers : 21 ; 3 × 7 ; 12 ; 2 3 c Show squares in factors in the following numbers : 36 ; 50 ; 20 d Deduce the simplified forms of the following numbers : 36 ; 50 ; 20 E.755 Knowing that 196 = 14 , give the exact value of the following numbers : a 1.96 b 19600 c 0.0196 d 1960000 E.726 Simplify the following radicals by writing them in the form a b with a and b integer where b is as small as possible. a 3 5 × 2 6 b 5 2 × 2 2 c 3 6 × 4 3 d 24 × 6 E.729 Reduce the following expressions to the form : a + b c with c as small as possible : a 2 × 3 × 7 b 2 + 3 7 c 2 5 × 3 × 10 d 8 18 3 2 e 18 − 2 f 8 × 2 − 2 2 g 2 2 − 3 h 5 − 1 20 + 5 i 5 − 1 2 j 7 + 2 2 k 5 + 1 5 − 1 https://chingmath.fr chapExoCorrec/5688 sacados/5688 4cm2 chapExoCorrec/742 sacados/742 chapExoCorrec/238 sacados/238 chapExoCorrec/272 sacados/272 chapExoCorrec/237 sacados/237 chapExoCorrec/769 sacados/769 chapExoCorrec/749 sacados/749 chapExoCorrec/755 sacados/755 chapExoCorrec/726 sacados/726 chapExoCorrec/729 sacados/729
3. Additive simplifications E.3909 Write B and C in the form, a b , with a and b integers ( b being the smallest possible) . B = 3 5 − 2 45 + 500 ; C = 3 + 4 2 − 19 4. Simplifications E.777 Simplify the following calculations as much as possible : A = 63 + 4 28 − 9 175 ; B = 15 × 35 7 C = − 150 3 + 2 8 − 7 72 E.795 Write all expressions in the form a b with b as small an integer as possible : a 75 b 48 16 c 3 + 5 3 d 2 + 8 e 5 2 × 4 18 f 7 6 − 3 24 g 27 + 3 12 − 5 75 E.796 Give the following results in the form a b where a is a relative integer and b is the smallest possi-ble positive integer. a 6 × 42 b 14 × 6 21 c 75 − 4 12 + 27 d 2 + 2 2 E.761 Write all expressions in the form a b with b as small an integer as possible : a 75 b 48 16 c 3 + 5 3 d 2 + 8 e 5 2 × 4 18 f 27 + 3 12 − 5 75 g 7 6 − 3 24 h 10 × 18 + 3 5 5. Number comparison E.292 Compare without the aid of a calculator: a 6 et 33 b 6 × 5 et 6 c 10 10 et 30 d 1 4 et 1 √ 15 e 5 + 3 et 5 + 3 f 2 2 − 3 et 17 − 12 2 E.1934 Without the aid of a calculator, carry out a comparison of the following pairs of numbers : a 2 19 et 5 3 b 1 35 et 1 6 c 1 2 − 3 et 3 + 2 d 12 − 7 et 5 e 3 5 + 2 et 47 + 6 10 f 6 11 × 3 × 4 7 3 12 et 2 50 E.295 Compare the following real numbers and explain your answer: a 32 5 et 15 14 b 1 2 3 et 1 3 2 c 3 + 2 et 2 6 + 4 d 3+ ı 3 et 4+ ı 4 e 3 − 12 et 19 − 12 3 f ( n − 1) 2 et n 2 − 1 pour n 1 g 2 n n + 1 et 3 n − 1 n pour n 1 E.325 This exercise must be completed without the aid of a calculator: 1 Consider the following two numbers : A = 2 3 ; B = 3 2 a Determine the exact values of A 2 and B 2 . b Compare the two numbers A and B . 2 Compare each pair of numbers below : a 6 et 3 6 b 1 2 3 et 1 3 2 c 3 2 et 3 d 2 3 et 2 − 8 https://chingmath.fr chapExoCorrec/3909 sacados/3909 chapExoCorrec/777 sacados/777 chapExoCorrec/795 sacados/795 chapExoCorrec/796 sacados/796 chapExoCorrec/761 sacados/761 chapExoCorrec/292 sacados/292 chapExoCorrec/1934 sacados/1934 chapExoCorrec/295 sacados/295 chapExoCorrec/325 sacados/325
E.342 Without using a calculator, compare the pairs of numbers below : a 45 and 7 b − 3 and − 57 c 2 3 and 3 2 d 10 5 and 5 10 e 1 4 and 1 ı f − 1 3 and − 1 3 E.345 Without the aid of a calculator, compare the following pairs of numbers, justifying your approach : A 7 2 ; 9 b − 1 ı ; − 1 4 c 1 8 ; 1 3 d 2 7 ; 5+ 3 E.328 During this exercise, the use of a calcula-tor is forbidden : 1 a Compare the two numbers 9 and 5 3 . b Which of these two spellings defines a number: 5 3 − 9 ; 9 − 5 3 2 Which of the writings below defines a number? a 80 − 9 b 5 − 2 6 c 5 − 3 3 E.4866 Compare the following pairs of numbers : a 2 √ 3 3 ; 3 √ 2 4 b 3 √ 5 5 ; 5 √ 3 6 c 5 2 ; 4 √ 10 5 d 5 √ 8 4 ; 8 √ 5 5 E.786 This exercise proposes to compare 5 and 3+ 2 . 1 a Draw a triangle AIB isosceles rectangle at I such that AI =3 cm . b Determine the length of segment [ AB ] . c Draw the point C such that ABC is rectangular at B and BC =3 . Show that : AC =3 3 2 Draw a half-right [ Ox ) and plot the lengths on this half-right so that : the point G such that : OG =3 √ 2 The point H such that : OH =3 3 The point J such that : OJ =3 2+3 3 3 Noting that : 9 × 5 = 9 × 4 + 9 × 1 . Draw a right-angled triangle whose hypotenuse has a length of 3 5 . 4 a On the half-right [ Ox ) , place the point K such that : OK = 3 5 b Graphically, compare the lengths OK and OJ . Deducts a comparison of the numbers 5 and 3+ 2 ? 5 Expand then simplify: 3+ 2 2 . Does this agree with the question 4 c . E.2848 Compare the following numbers, justify-ing your method : a 1 46 et 1 3 5 b 3 3 − 2 2 et 35 + 12 6 c 2 + 3 1 − 2 et 2 − 3 1 + 2 d 3 4 × 12 2 3 8 × 4 4 et 1 36 6. Simplifications E.741 madagascar - June 2006 Consider the numbers : C = 5 3 + 2 27 ; D = 3 2 × 6 Write the numbers C and D in the form a 3 , a being an integer. E.236 Perform the following operations, putting the result in the following form p √ q where p is a relative in-teger and q is a natural number as small as possible : a 500 b 252 c 6 × 48 d 3 2 +4 2 e 5 3+2 75 − 3 12 f 10+ 2 · 5 5 E.4378 Give the simplified form of each of the following expressions : a 7 500 b 50 × 48 c 45+2 500 − 80 d 2 75 − 48 e 15 × 10 4 16 × 10 − 6 f 98 × 6 E.4420 Simplify the following calculations : a 4 2 + 6 × 48 b 27 + 2 12 − 5 75 c 75 + 12 2 3 d 8 × 75 5 ×√ 90 + 24 × 12 7. Square root, fractions, power E.743 We give : A = 7 3 − 2 3 ÷ 8 7 ; B = 12 − 7 3 − 75 C = 0.3 × 10 2 × 5 × 10 − 3 4 × 10 − 4 1 Calculate A and give the result as an irreducible fraction. 2 Write B in the form a b , where a is a relative integer https://chingmath.fr chapExoCorrec/342 sacados/342 chapExoCorrec/345 sacados/345 chapExoCorrec/328 sacados/328 chapExoCorrec/4866 sacados/4866 chapExoCorrec/786 sacados/786 chapExoCorrec/2848 sacados/2848 chapExoCorrec/741 sacados/741 Madagascar - Juin 2006 chapExoCorrec/236 sacados/236 chapExoCorrec/4378 sacados/4378 chapExoCorrec/4420 sacados/4420 chapExoCorrec/743 sacados/743 Dijon - Juin 2006
and b is the smallest possible natural number. 3 Calculate C and give its scientific form E.767 Intermediate calculations must be shown on the copy 1 Write in the form a 3 , a being an integer, the number: A = 75 + 4 12 2 Prove that : a 2 + 3 4 3 4 − 5 = − 11 17 b 35 × 10 22 × 2 × 10 − 2 6 42 × 10 10 = 5 3 E.2334 We give the numbers : A = 3 7 − 2 7 × 21 8 ; B = 3 × 10 2 × 1.8 × 10 − 3 6 × 10 4 C = 12 − 5 75 + 2 147 1 Calculate A and give the result as an irreducible fraction. Write down all the steps in the calculation. 2 a Give the decimal form of B . b Express B in scientific writing. 3 Write C in the form a 3 , where a is an integer. E.2335 1 Write A as an irreducible fraction : A = 4 3 − 1 7 6 − 2 2 We give : B = 4 × 10 − 2 × 9 × 10 6 6 × 10 7 × 10 2 × 10 3 2 Give the scientific form of B . 3 Write C in the form a 6 where a is a relative integer: C = 96 + 5 6 − 3 150 E.787 Calculate the following expressions. The result will be given as an integer. Intermediate calculations will appear on the copy A = 96 × 10 − 4 × 5 × 10 − 2 3 × 10 − 1 × 2 × 10 − 6 ; B = 11 ÷ 2 3 − 5 2 C = 2 3 − 3 2 3 + 3 E.790 1 We give : A = 2 3 +3 1 3 +5 Write A as an irreducible fraction. 2 We give : B =2 50 − 3 8+7 18 Write B in the form a 2 , with a an integer. 3 We give : C = 2.6 × 10 2 × 1.7 × 10 2 0.2 × 10 5 × 10 3 Give the scientific writing of C . E.797 1 Showing the steps, calculate and give the scientific writ-ing of : D = 2 × 10 3 × 5 × (10 − 5 ) 2 2 + 18 2 a E = 2 ×√ 27 + √ 18 ×√ 6 Calculate and write E in the form a 3 (where a rela-tive integer) b F = 2 − 4 2 + 4 2 Calculate and write F in the form b 2 (where b rela-tive integer) E.627 1 Consider the following two expressions : A = 3 5 − 1 2 × 5 2 ; B = 16 × 10 − 1 × 2 10 3 2 × 10 − 8 × 80 a Calculate A and express the result as an irreducible fraction. b Verify that B is an integer Write out the steps of the calculation. c Brice claims that ˇ A is the opposite of B .ı Is this true? Explain your answer. 2 Consider the following two expressions : C = 2 24+ 96 − 600 ; D = 3 − 2 3+5 2 a Express C in the form a 6 , where a is a relative inte-ger b Expand and simplify D . E.752 1 Consider the number: A = 1 7 + 6 7 ÷ 12 35 Calculate A , detailing the steps in the calculation, and write the result as an irreducible fraction. 2 Consider the numbers : B = 17 − 1 17+1 ; C = 3 − 7 2 ; D = B − C a Expand and reduce B and C . b Write D in the form a 7 , where a denotes an integer. E.753 1 We give : A = 3 7 − 15 7 ÷ 5 24 . Calculate A and give the result as an irreducible fraction. 2 We give : B = 300 − 4 27 + 6 3 ; C = 5 + 3 2 D = 2 + 5 2 − 5 a Write B in the form b 3 , where b is an integer. b Write C in the form e + f 3 , with e and f integers. c Show that D is an integer. https://chingmath.fr chapExoCorrec/767 sacados/767 Groupe Ouest - Septembre 2002 - 4 points chapExoCorrec/2334 sacados/2334 Brevet 2008 chapExoCorrec/2335 sacados/2335 chapExoCorrec/787 sacados/787 Groupe Ouest - 2004 - 3 points chapExoCorrec/790 sacados/790 chapExoCorrec/797 sacados/797 chapExoCorrec/627 sacados/627 chapExoCorrec/752 sacados/752 chapExoCorrec/753 sacados/753 Groupe Sud - Juin 2004 - 4,5 points
E.4393 Perform the following calculations and give the results in simplified form : a 25 − 5 × 2 5 − 2 × 10 b 4 − 8 3 1 5 − 5 c 15 × 10 − 3 × 6 × 10 5 5 × 10 5 × 3 × 10 − 7 2 d 4 3 × 3 2 × 10 15 2 × 2 5 e 75 + 2 27 − 8 × 6 f 24 + 27 × 8 5 8 − 2 18 E.4426 Establish the following equalities: a 3 − 2 3 3 2 − 4 12 = 2 b 3 − 3 × 2 1 + 1 2 − 5 − 4 × 2 2 7 − 2 = − 15 4 c 27 + 12 48 × 36 = 5 24 d (3 × 15) 10 × 6 4 5 8 × 12 12 × 3 3 =2 − 20 × 3 9 × 5 2 e 24 × 75+ 50=35 2 f 3 × 10 5 × 21 × 10 2 15 × 10 − 2 × 3 × 10 5 =1.4 × 10 4 E.262 1 Perform the following calculation: A = 1 + 2 1 + 2 3 + 1 3 2 Write the following quotient in the form 2 m × 3 n × 5 p × 7 q where m , n , p , q are relative integers : B = 3 × 15 2 × 2 × 5 3 ) − 2 7 3 × 12 4 3 Simplify the writing of the following expressions : C = 3 2 + 5 2 3 6 − 2 2 ; D = 1 − 6 1 + 6 E.2328 We give : E = 2 3 + 17 2 × 4 3 ; F = 6 × 3 × 16 2 1 Demonstrate that the numbers E and F are equal. 2 We give G = 10 − 1 + a × 10 2 . Calculate the number a for the equality E = G to be true. 8. Conjugated expressions E.8362 1 Expand the expression : 2+1 2 − 1 2 Use the previous question to simplify the expression : 1 + 1 2 + 1 E.268 1 a Show that 2+ 3 2 − 3 is an integer. b To simplify writing the quotient 2 3 2+ 3 , we’ll multiply its numerator and denominator by 2 − 3 . Note that the denominator then has an integer value. 2 a Show that 2+3 5 2 − 3 5 is a relative integer. b Use this result to write 5 − 2 2 − 3 5 with an integer de-nominator. 3 Write 4 5 − 1 with an integer denominator. 4 Write 3 − 2 3+ 2 with an integer denominator. E.4980 Write the expressions below without square roots in the denominator : a 2 2 − 1 b 3 3 + 1 c 2 2 6 − 4 d 2 2 − 3 E.4981 Write the following expressions without square roots in the denominator : a 2 x x − 1 b x + x x − x c x 2 x + 2 x for x =4 E.232 Demonstrate the following equalities: 1 3 + 2 3 − 2 + 3 − 2 3 + 2 = 10 2 4 − 2 3 = 3 − 1 Find the simplified expression of 3 − 1 2 3 x − y x − y = x − y x + y with x ∈ R + ∗ and y ∈ R + ∗ such as : x = y . E.5031 Établir l’égalité: 1 1 + 2 + 1 2 + 3 + 1 3 + 2 = 1 E.258 Simplify the writing of each of the follow-ing expressions : a √ 63 − 5 7 + 2 √ 2800 b 2 3+4 2 2 − 3 c 2 + √ 24 6 d 5 − 2 1 − 2 E.1779 Perform the following calculations and give the result in simplified form : a 4800 − 2 75 + 2 × 54 b 6 + 72 3 2 − 24 c 2 2 − 3 7 d 1 − 7 14 − 4 + 2 https://chingmath.fr chapExoCorrec/4393 sacados/4393 chapExoCorrec/4426 sacados/4426 chapExoCorrec/262 sacados/262 chapExoCorrec/2328 sacados/2328 France - 2008 sacados/8362 chapExoCorrec/268 sacados/268 chapExoCorrec/4980 sacados/4980 chapExoCorrec/4981 sacados/4981 chapExoCorrec/232 sacados/232 chapExoCorrec/5031 sacados/5031 chapExoCorrec/258 sacados/258 chapExoCorrec/1779 sacados/1779
E.281 Perform the following calculations and give their results in simplified form. a 150 + 3 600 − 7 294 b 2 18 3 − 2 c 2 − 3 3 3 + 2 2 d 28 − 3 7 e 2 + 5 1 − 5 f 1 + 3 3 − 2 E.239 Simplify the following entries : a 175 − 10 112 + 7 b 2 2 − 2 200 + 98 + 18 c 3 3 − 6 3 d 27 − 5 3 − 2 e 3 − 2 3 + 2 E.261 1 Show that : 3+ 6 3 − 6 − 6 3 12 =0 2 Write 7 4 − 3 without root in denominator 3 Perform the following calculation: 6+2 3 − 2 4 Establish the following equality: 2 6 − 2 − 3= 2 E.294 Compare without using a calculator: a 152 240 et 110 b 7 21 et 1 3 c 23 15 et 23 13 d 12 25 et 77 125 e 3 + 2 3 et 2 + 2 2 f 2 − 1 et 1 2 + 1 g 2 + 3 et 5 + 2 6 E.355 Compare the following numbers, justify-ing your method : a 8 2 et 7 3 7 b 3 + 5 et 8 + 2 15 c 1 1 + 2 et 2 − 1 d 15 − 2 14 et 14 − 2 15 E.327 Compare the following numbers and ex-plain your reasoning : a 7 8 et 8 7 b 1 3 5 et 1 5 3 c 11+ 2 12 et 12+ 2 13 d 4+ 2 et 6 + 2 8 e 3 − 2 et 11 − 6 2 f 1 1 − 6 et 1 + √ 6 9. Square root and PGCD E.5320 1 Determine the PGCD of the two integers 1323 and 243 . 2 Give the simplified form of the fraction 1323 243 E.5321 1 Determine the PGCD of the two integers 343 and 175 . 2 Give the simplified form of the fraction 343 175 E.5322 1 Determine the PGCD of the two integers 847 and 63 . 2 Write the number A in the form a b : A = 847 + 63 E.5323 1 Determine the PGCD of the two integers 567 and 175 . 2 Write the number A in the form a b : A = 567 + 175 E.773 1 Without calculating their PGCD , say why the integers 648 and 972 are not prime to each other. 2 a Calculate pgcd (972 ; 648) . b Prove that : 648+ 972=18 3+ 2 E.2131 1 Without any calculation explain why we can simplify the fraction 4 114 7 650 . 2 Calculate the PGCD of the integers 4 114 and 7 650 using the method of your choice, giving details of the calcula-tions. 3 Make the fraction 4 114 7 650 irreducible, specifying by which number you simplify. 4 Using the results of the previous questions, put the fol-lowing expression A into the form n 34 , where n is a relative integer, detailing the calculations : 5 4 114 − 4 7 650 https://chingmath.fr chapExoCorrec/281 sacados/281 chapExoCorrec/239 sacados/239 chapExoCorrec/261 sacados/261 chapExoCorrec/294 sacados/294 chapExoCorrec/355 sacados/355 chapExoCorrec/327 sacados/327 chapExoCorrec/5320 sacados/5320 chapExoCorrec/5321 sacados/5321 chapExoCorrec/5322 sacados/5322 chapExoCorrec/5323 sacados/5323 chapExoCorrec/773 sacados/773 Afrique - Juin 2006 chapExoCorrec/2131 sacados/2131 Asie - Juin 2008 - 3 points