Grade 10 / Square roots 63 exercises (100% corrected)

a
9cm21cm1cm 16cm27;5cm 4cm2 AO1cmBCDEFGHIJKLMNPQR ChingQuizz : 6 exercises available for Quizz assessment : 1. Problem situations E.5783 Below are two squares, some of whose measurements are shown in the figure. Is there a square whose area is the sum of the areas of the two squares shown? If so, give the measure of its side. E.5784 Opposite are two squares, some of whose measurements are specified in figure : Is there a square whose area is the sum of the areas of the two squares shown? If so, give the measure of its side. E.5785 Opposite are two squares, some of whose measure-ments are specified in fig-ure : Is there a square whose area is the sum of the areas of the two squares shown? If so, give the measurement of its side. 2. Introduction E.748 The figure below is constructed as follows : The triangle OAB is isosceles at A such that OA =1 cm ; Outside the triangle OAB and on the hypotenuse [ OB ] , we construct a right-angled triangle at B such that : BC =1 ; and so on. . . 1 a Justify the following two equalities: OB 2 = 2 ; OC 2 = 3 b Using a calculator, give an approximate value for the lengths OB and OC to the nearest 10 − 3 . 2 a Justify that : OD =2 cm . b Briefly justify that : OI =3 cm . E.792 Answer the following question using the calculator: 1 Give the truncation of the following numbers to the near-est hundredth : a 2 b 3 c 10 2 Give the rounded value of the following numbers to the nearest 10 − 2 : a 52 b 4 + 0.03 c 72 + 2 2 3 Determine the exact value of the following numbers : a 4 b 25 + 75 c 0.01 4 Solve the following equations (each of these equations ad-mit two solutions) : a x 2 = 9 b x 2 = 100 c x 2 = 2 E.766 Without the aid of a calculator, give the exact value of each of the square roots below : a 4 b 400 c 20 + 44 d 0.49 e 121 f 0.25 https://chingmath.fr chapExoCorrec/5783 sacados/5783 9cm21cm1cm chapExoCorrec/5784 sacados/5784 16cm27;5cm chapExoCorrec/5785 sacados/5785 4cm2 chapExoCorrec/748 sacados/748 AO1cmBCDEFGHIJKLMNPQR chapExoCorrec/792 sacados/792 chapExoCorrec/766 sacados/766
6√5cm2√30cm2√15cmABC E.756 Without the aid of the calculator, justify that none of the expressions below make sense : − 4 ; − 1 ; 5 − 9 3. Around the definition E.252 1 a Define the number 3 . b Explain why the notation − 1 does not define any real number. 2 Some of the expressions below do not define a number. Which ones? a 5 b ( − 2) 2 c − 16 d 3 e 5 + 6 f 13 − 136 E.784 Let a be a positive number, the defi-nition of the square root allows me to establish the following two relationships : a 2 = a ; a 2 = a . Use these two properties to simplify, if possible, the following expressions : a 4 2 + 6 2 b 3 2 + 4 2 c 4 + 6 2 d (4 + 6) 2 e 4 2 × 6 2 f 4 2 × 6 2 E.757 1 a Give the value of the following expressions : 9 + 16 ; 9 + 16 b Give the value of the following expressions : 169 − 25 ; 169 − 25 2 What can be said about the relationships below : a a + b = a + b b a − b = a − b E.754 1 Is the triangle ABC below right-angled? (On the figure the dimensions are really not respected) 2 Give the value of the following number: D = 6 + 6 + 6 + 6 + 6 + 6 + 3 + 3 E.246 Justify the following two equalities: 3 4 = 3 ; 7 10 = 7 5 4. First equations of the second degree E.788 1 Determine two values of x verifying equality: x 2 = 4 . 2 Quel (s) nombre (s) vérifie (nt) equality: x 2 = 0 . 3 Is there a number x verifying the equality: x 2 = − 1 . Justify your answer. E.765 1 Give the two solution numbers of the equation x 2 =4 . 2 Solve the following equations : a x 2 = 0 b x 2 = − 1 c ( x + 1) 2 = 0 d ( x − 1) 2 = 4 5. Multiplicative relations and simplifications E.1955 1 a Calculate the square of the number 5 × 3 . b Compare the two numbers : 5 × 3 et 15 c For any number a and b positive, justify the equality: a × b = a × b 2 a Determine two integers a and b such that : 50 = a 2 × b b Establish equality: 50 = 5 2 https://chingmath.fr chapExoCorrec/756 sacados/756 chapExoCorrec/252 sacados/252 chapExoCorrec/784 sacados/784 chapExoCorrec/757 sacados/757 chapExoCorrec/754 sacados/754 6√5cm2√30cm2√15cmABC chapExoCorrec/246 sacados/246 chapExoCorrec/788 sacados/788 chapExoCorrec/765 sacados/765 chapExoCorrec/1955 sacados/1955
E.751 Proposition: for any number a and b positive or zero, we have the equality: a × b = a × b 1 Write each of the numbers below in product form, where the maximum factors are numbers squared (example 50=5 2 × 2 ) a 75 b 32 c 18 d 72 e 1 000 f 242 2 Give a simplified writing of the following square roots : a 75 b 32 c 18 d 72 e 1 000 f 242 E.782 Write the following radicals in the form a b with a and b two integers where b is as small as possible : a 3 2 × 2 b 13 × 4 2 c 12 d 48 e 1 600 f 360 E.771 Simplify the following radicals : a 4 b 84 c 200 d 30 + 42 e 98 f 150 g 0.01 h 0.36 E.8390 Simplify the expression for each of the following products : a 6 × 40 b 3 × 15 c 8 × 18 E.750 Write the following calculations in the form a b where a and b are integers with b as small as possible : a 5 × 30 b 24 × 6 c 5 2 × 2 2 d 3 6 × 4 3 e 39 × 2 13 f 2 15 2 6. Multiplicative relations and quotient E.8324 1 Simplifier le calcul suivant : 7 3 × 7 3 2 a Complete the following sentence : 7 3 is a number whose square is . . . . . . b Complete the equality: 7 3 = : : : : : : 3 For all positive numbers a and b , with b =0 , justify the equality: a b = a b E.791 Proposition: let a and b be two positive numbers with b =0 . We have the relation: a b = a b Write the following calculations in the form a b where a and b are integers with b as small as possible : a 2 × 5 2 b 30 6 × 2 5 c 15 14 × 35 6 E.275 Justify that each of the expressions shown below represents the inverse of the number 8 3 : a 3 8 b 3 8 8 c 18 16 7. Simplifications of quotients with radicals in the denominator E.250 Using the following example, write the following quotients without radicals in the denominator, and simplify the quotients as much as possible. 2 6 = 2 × 6 6 × 6 = 2 6 6 = 6 3 a 5 2 10 b 6 3 5 c 2 + 1 6 d 21 70 E.1742 1 Justify each of the following equalities: a 3 7 × 7 7 = 3 7 7 b 4 2 × 2 2 = 2 2 c 1 3 = 3 3 2 Using the previous question, establish the following equalities: https://chingmath.fr chapExoCorrec/751 sacados/751 chapExoCorrec/782 sacados/782 chapExoCorrec/771 sacados/771 chapExoCorrec/8390 sacados/8390 chapExoCorrec/750 sacados/750 chapExoCorrec/8324 sacados/8324 chapExoCorrec/791 sacados/791 chapExoCorrec/275 sacados/275 chapExoCorrec/250 sacados/250 chapExoCorrec/1742 sacados/1742
ABC√27√12 ABCMN√6√262√3 a 2 3 = 6 3 b 7 + 3 7 = 1 + 3 7 7 c 4 2 + 2 = 3 2 E.733 Write the following fractions without a radical in the denominator : a 1 3 b 3 2 c 28 7 E.775 Simplify the expressions below with-out a radical in the denominator : a 2 2 b 3 2 c 5 15 d 2 18 e 27 3 8. Multiplicative relations and problems E.783 1 Expand : A ( x )=(2 x +1)(2 x − 1) . 2 Calculate A ( x ) for x = 5 . 3 Explain how the first question can be used to calculate: 20 001 × 19 999 E.2352 Given the following expression : K =(5 x − 3) 2 +6(5 x − 3) 1 Expand K and give its simplified expression. 2 Give the factored expression of K . 3 Give a simplified form of K when x = 2 . E.3833 Consider the triangle ABC shown below, some of whose side measures are plotted on figure : Determine the area of this triangle. E.3856 Consider the expression : A =(2 x − 3)( x − 4) − (2 x − 3) 2 1 Expand and reduce A . 2 Calculate A when x = 3 2 , then when x =3 · √ 2 3 Factor A . E.5239 Consider the triangle ABC où M is a point of [ AB ] and N is a point of [ AC ] . We have the following measurements : AB = 6 ; AC = 2 3 ; AM = 6 ; AN = 2 Show that the straight lines ( BC ) and ( MN ) are parallel. E.249 After a quiz, three students discuss their results ; here are their answers to the question : ˇ Give the inverse of √ 2 √ 3 ı Student A answered 3 2 ; Student’s answer B was 3 6 ; Student C wrote 1 2 × 6 on his copy. Using the definition of the inverse of a number , show that all three answers are correct. 9. Additive simplifications E.3832 1 Simplify the writing of the sum below : A = 2 + 2 2 2 a Simplify the expression of the following square roots : 50 ; 32 b Deduce from the previous question a simplification of the sum : B = 50 + 32 + 2 3 Consider the number: C =2 27+5 75 Justify the following simplification : C =31 3 https://chingmath.fr chapExoCorrec/733 sacados/733 chapExoCorrec/775 sacados/775 chapExoCorrec/783 sacados/783 chapExoCorrec/2352 sacados/2352 chapExoCorrec/3833 sacados/3833 ABC√27√12 chapExoCorrec/3856 sacados/3856 Extrait de Reims Septembre 2002 chapExoCorrec/5239 sacados/5239 ABCMN√6√262√3 chapExoCorrec/249 sacados/249 chapExoCorrec/3832 sacados/3832
TCERPI3532 E.785 Simplify as much as possible the writ-ing of the following calculations : a 3 + 2 3 b 12 + 3 c 3 × 6 + 2 d 8 + 2 E.734 Give the expressions below in the form a b with a and b two integers where b is as small as possible : a 3 + 3 b 2 5 + 3 5 c 2 − 4 2 d 8 + 2 e 27 − 8 3 f 50 − 72 E.728 1 Reduce each of the expressions below to the form a √ b where b is an integer: a 2 + 2 2 b 2 8 c 3 50 + 2 2 Do the same with the following expressions : a 2 18 + 50 b 4 12 − 2 75 c 5 + 3 2 × 40 − 45 E.772 Write in the form a b with a and b integers. a 3 28 − 9 7 b 2 + 32 + 200 c 2 45 − 3 5 + 20 d 4500 + 3 5 − 2 125 E.758 1 Write the following calculation in the form a 3 where a is an integer: 2 48 + 7 3 − 75 2 Show that A is an integer: 63 − 4 2 + 18 × 2 + 2 8 − 3 7 E.774 The unit of length is the cen-timeter. RECT is a rectangle. 1 Calculate the perimeter of the triangle TIP . 2 Two students calculated the perimeter of the triangle TIP . Marcel found : 2 17 2+1 . Paul found : 2 34 1+ 1 2 . a Is Marcel’s answer correct? b Is Paul’s answer correct? 10. Simple distributivity and square root E.759 Expand and simplify the expressions below : a 1 + 5 15 b 3 + 2 2 c 1 + 3 6 2 d 3 2 5 − 2 E.9573 Expand and simplify the following expressions : a 2 18 + 2 b 5 5 − 45 E.9575 Calculate and simplify the following roots a 3 + 2 2 b 5 × 2 15 − 3 5) 11. Double distributivity and square roots E.780 Expand and give the result in simpli-fied form : a 2 + 3 1 − 2 b 2 + 3 3 − 2 c 5 + 2 5 + 2 d 3 × 2 2 e 2 − 5 5 + 2 E.736 Calculate and simplify the following radical expressions as much as possible : a 2 + 3 1 − 2 b 2 − 5 2 + 5 + 1 E.9574 Calculate and simplify the following calculations : 2 − 5 2 + 5 + 1 E.744 Give the result of the following cal-culations in the form a + b c , where a and b are relative numbers and where c is the smallest possible positive integer. a 2 − 3 2+ 3 b 4 5 − 3 3 3+2 5 https://chingmath.fr chapExoCorrec/785 sacados/785 chapExoCorrec/734 sacados/734 chapExoCorrec/728 sacados/728 chapExoCorrec/772 sacados/772 chapExoCorrec/758 sacados/758 chapExoCorrec/774 sacados/774 Brevet 91 TCERPI3532 chapExoCorrec/759 sacados/759 chapExoCorrec/9573 sacados/9573 chapExoCorrec/9575 sacados/9575 chapExoCorrec/780 sacados/780 chapExoCorrec/736 sacados/736 chapExoCorrec/9574 sacados/9574 chapExoCorrec/744 sacados/744
33√27236 ABCba E.9576 Calculate and simplify as far as pos-sible the following roots : a 2 − 3 2 b 3 5 + 2 2 E.9577 Expand and simplify the following expressions : a 2 3 + 1 2 b 3 5 + 2 2 E.267 1 Show that the following two numbers are inverses of each other : 5 + 1 2 ; 5 − 1 2 2 Show that : 5+1 2 2 =1+ 5+1 2 12. Algebra and problems E.760 In this exercise, all lengths are given in cm . Consider the two figures below, where : The length of one side of the square is 3+3 The dimensions of the rectangle are 72+3 6 and 2 . 1 Calculate the area A of the square ; simplify the expres-sion obtained. 2 Calculate the area A of the rectangle. 3 Check that this rectangle and this square have the same area. E.264 Consider the triangle ABC with the following measures : AB = 5 − 2 ; BC = 5 − 1 ; AC = 2 × 5 Is the triangle ABC right-angled? Justify. E.2329 1 Verify that − 2 3 is solution of the following equation : ( E ) : 3 x 2 + 5 x + 2 = 0 2 Verify that 2 − 1 is a solution to the following equation : ( F ) : x 2 + 2 x − 1 = 0 13. Square root and absolute value E.8323 1 Complete the table below : x − 5 − 3 0 1 2 5 x 2 d (0 ; x ) 2 For any real number x ( x ∈ R ) , what can we say about x 2 and d (0 ; x ) ? 14. Triangular inequality E.8363 Let a and b be two strictly posi-tive real numbers. Consider the triangle ABC right-angled A whose sides [ AB ] and [ AC ] have the measures a and b respectively: 1 Express the length of the hypotenuse [ BC ] in terms of the numbers a and b . 2 Which property asserts the inequality: a + b a + b for any positive or zero real number a and b ? https://chingmath.fr chapExoCorrec/9576 sacados/9576 chapExoCorrec/9577 sacados/9577 chapExoCorrec/267 sacados/267 chapExoCorrec/760 sacados/760 33√27236 chapExoCorrec/264 sacados/264 chapExoCorrec/2329 sacados/2329 chapExoCorrec/8323 sacados/8323 chapExoCorrec/8363 sacados/8363 ABCba
1aABCD 1aABCD hxvantellesradierAmontAvalPortesEcluse 15. In-depth study: extraction of the square root in geometry E.9722 Let a be a real number strictly greater than 1 . Consider a segment [ AB ] and a point C be-longing to [ AB ] such that : AC = 1 ; CB = a We construct the halfcircle C of diameter [ AB ] and the point D intersection of C with the straight line passing through C and perpendicular to ( AB ) . Show that : CD = a E.9723 Let a be a real number strictly greater than 1 . Consider a segment [ AB ] and a point C be-longing to [ AB ] such that : AC = 1 ; AB = a We construct the semicircle C with diameter [ AB ] and the point D , the intersection of C with the line passing through C and perpendicular to ( AB ) . Prove that : AD = a Note: this method of construction using a ruler and com-pass allows us to obtain a segment of length √ a . 16. Unclassified exercises E.2468 Using the decomposition of integers into products of prime factors, write each of the following numbers in the form p q where p ∈ N and q ∈ N and where q is as small as possible. a 432 b 126 c 42 E.6278 We will take a closer look at how a lock is filled to allow a barge to pass from upstream to downstream. Principle : The water level in the lock is raised to the level of the upstream canal so that the barge can then pass through the lock. The lock is then emptied and the water level drops to that of the downstream canal. The barge can leave the lock and continue downstream. All length measurements are expressed in meters. Note h the height of the water level upstream and x the height of the water level in the lock. These heights are measured from the lock’s bed (bottom) . (see diagram above) . When the barge arrives at the lock, we have : h =4.3 m ; x =1.8 m The speed of the water flowing through the sluice gate (gate) is given by the following formula : v = 2 g h − x where g =9.81 (acceleration in meters per second squared, de- https://chingmath.fr chapExoCorrec/9722 sacados/9722 1aABCD chapExoCorrec/9723 sacados/9723 1aABCD chapExoCorrec/2468 sacados/2468 chapExoCorrec/6278 sacados/6278 hxvantellesradierAmontAvalPortesEcluse
hauteur(m0123456vitesse(m=s123456789 noted by m · s − 2 ) . 1 Calculate the speed of the water flowing through the sluice gate at the moment it opens, rounded to the near-est m = s . (Opening is considered to be instantaneous) . 2 For what value of x will the water flow velocity be zero? What does this mean for the water level in the lock? 3 The graph given below represents the flow velocity of the water through the sluice as a function of the level x of the water in the lock Determine, by graphical reading, the flow velocity when the height of the water in the lock is 3.4 m . E.793 Write the following expression in the form a b with b an integer as small as possible : 27 + 3 12 − 5 75 E.11624 Montrer que 3 − 2 est une solution de l’équation: x 2 + 4 x + 1 = 0 E.11625 Montrer que 2 − 2 2 est une solution de l’équation: 2 x 2 + 4 x + 1 = 0 E.11626 Montrer que le nombre 2 3 − 3 3 est une solution de l’équation: 3 x 2 + 6 x − 1 = 0 https://chingmath.fr hauteur(m0123456vitesse(m=s123456789 chapExoCorrec/793 sacados/793 chapExoCorrec/11624 sacados/11624 chapExoCorrec/11625 sacados/11625 chapExoCorrec/11626 sacados/11626