##### Natural Science Mathematics Question Bank for Entrance Exam(2365 Questions)
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# Unit-1: The Number System

Unit-1: The Number System

1. Which of the following is a prime number?
A. 87
B. 151
C. 161
D. 171

2. Which of the following expressions gives the prime factorization of 1008?
A. $${2^2}{\rm{ \times }}{3^2}{\rm{ \times }}11$$
B. $${2^3} \times {3^3} \times 7$$
C. $${2^4} \times {3^2} \times 7$$
D. $${2^3} \times {3^2} \times 13$$

3. If the prime factorization of natural number is then how many factors does a number have?
A. 5
B. 6
C. 7
D. 8

4. Which of the following is the fractional form of the decimal number  $$5.\overline {575}$$?
A. $$\frac{{5570}}{{999}}$$
B. $$\frac{{5570}}{{900}}$$
C. $$\frac{{5575}}{{990}}$$
D. $$\frac{{5570}}{{990}}$$

5. The number 3.765765…. is equal to:
A. $$\frac{{418}}{{111}}$$
B. $$\frac{{1255}}{{333}}$$
C. $$\frac{{418}}{{110}}$$
D. $$\frac{{1255}}{{310}}$$

6. Which of the following is the greatest common factor (GCF)and least common multiple(LCM)of 1800 and 756respectively?
A. 36and 37800
B. 72and 1512
C. 30 and 504
D. 210 and 5400

7. Which of the following numbers is  divisible by 6?
A. 126
B. 123
C. 999
D. 6361

8. What is the simplified form of the expression $$\sqrt{{\frac{1}{5}}}*\sqrt{{\frac{1}{9}*}}\sqrt{{\frac{5}{{27}}}}?$$
A. $$\frac{1}{5}$$
B. $$\frac{1}{3}$$
C. $$\frac{1}{4}$$
D. $$\frac{{\sqrt{3}}}{3}$$

9. On the number line, $$\sqrt {35} – 6$$ is located between —- and —- .
A. 0 and 1
B. 1 and 2
C. -1 and 0
D. -2 and -1

10. On the number line, $$\sqrt {28} + 5$$ is located between:
A. 9 & 10
B. 10 &11
C. 11& 12
D. 12 & 13

11. If x is an odd natural number such that LCM =1400, then x is equal to__
A. 35
B. 75
C. 125
D. 175

12. There are between 60 and 70 eggs in a basket. If one counts them out 3 at a time, he has 2 left over, but if he counts them out 5 at a time, he has 3 left over. How many eggs are there in the basket?
A. 66
B. 67
C. 68
D. 69

13. The GCF of two numbers is 12 and their LCM is 180. If one of the numbers is 36, then what is the other number?
A. 45
B. 60
C. 30
D. 120

14. When the expression  $$\frac{{\sqrt {5 – 2\sqrt 6 } .\sqrt {5 + 2\sqrt 6 } }}{{\sqrt{{ – 8}}}}$$  is simplified, it gives:

A. $$– \frac{{\sqrt {13} }}{2}$$
B. $$\frac{{\sqrt {13} }}{2}$$
C. $$– \frac{1}{2}$$
D. $$\frac{1}{2}$$

15. If $$\frac{a}{b}$$  and $$\frac{c}{d}$$ are two rational numbers such that $$\frac{a}{b}$$<$$\frac{c}{d}$$  , then which of the following is necessarily between  $$\frac{a}{b}$$and $$\frac{c}{d}$$?
A. $$\frac{c}{d}$$-$$\frac{a}{b}$$
B. $$\frac{{ad + bc}}{{2bd}}$$
C. $$\frac{{ab + cd}}{{bd}}$$
D. $$\frac{{ab + cd}}{{2bd}}$$

16. Which of the following is true?
A. $$\frac{{22}}{7}$$is an irrational number
B. $$\sqrt{{ – 3}}$$  is a real number
C.  $$\sqrt{{ – 2}}$$is an irrational number
D. $$\frac{{22}}{7}$$is an exact value of π

17. Which of the following is not a rational number?
A. $$\frac{{\sqrt {12} }}{{\sqrt 3 }}$$
B. $$\frac{{\sqrt {{4^{\frac{1}{2}}}} }}{{\sqrt 2 }}$$
C. $$\frac{{\sqrt 2 – 1}}{{1 – \sqrt 2 }}$$
D. 7.545454…
E. None

18. Which of the following is not true?
A. $$\sqrt {5 + 2\sqrt 6 }$$ =$$\sqrt 2 + \;\sqrt 3$$
B. $$\sqrt {5 – 2\sqrt 6 }$$is a rational number
C. $$\sqrt {9 – 2\sqrt {14} } < \sqrt 7 + \sqrt 2$$
D.  $$\sqrt {4 + \sqrt {15} } = \frac{{\sqrt 3 + \sqrt 5 }}{{\sqrt 2 }}$$

19. For any real numbers , which of the following is always true?
A. If $$a + b$$ is a rational number, then  are rational numbers.
B. If $$a + b$$ and b are rational numbers, then is a rational.
C. If $$\frac{a}{b}$$ is a rational number, then  are rational numbers.
D. If ab is an irrational number, then  are irrational numbers.

20. Which one of the following can never be a rational number?
A. The sum of two irrational numbers.
B. The sum of a rational and an irrational numbers.
C. The product of a rational and an irrational numbers.
D. The ratio of two irrational numbers.

21. For natural numbers p, q, x and y, if p is a factor of x and q is a factor of y, then which of the following statements is not correct?
A. $$pq$$is a factor of $$xy$$
B. $$p + q$$ is a factor of $$xy$$
C. p is a factor of$$xy$$
D. q is a factor of$$xy$$

22. Which of the following is not true?
A. If you write 0.006217 to 3 significant figures, it becomes 0.00622
B. If you write 0.014 to 2 decimal places, it becomes 0.01
C. If you write 48987 to 1 significant figures, it becomes 50,000
D. If you write 764.316 to 1 decimal places, it becomes 764.32

23. When the number 753,968,563 is rounded to the nearest ten thousands, it will be equal to:
A. 753,970,000
B. 754,000,000
C. 753,960,000
D. 753,964,000

24. A certain publishing enterprise printed 83,462 copies of mathematics student text book for grade 9. If you are asked on radio to report a number of these text books to the nearest 100, then what would be correct figure that you will report?
A. 83,000
B. 83,500
C. 84,000
D. 83,400

25. How many significant figures are there in 0.001050?
A. 5
B. 4
C. 6
D. 7

26. The scientific notation for 0.00023 is:
A. $$2.3 \times {10^{ – 3}}$$
B. $$2.3 \times {10^3}$$
C. $$2.3 \times {10^{ – 4}}$$
D. $$2.3 \times {10^4}$$

27.  $$\sqrt {64\sqrt{{{y^{12}}{z^6}}}} isequalto:$$
$$A.64{y^2}\left| z \right|$$
$$B.8{y^{2}}\left| z \right|$$
$$C.8{y^2}z$$
$$D.8{y^2}z$$

28. If  are real numbers and a>b>0, then what is the simplified form of $$\sqrt {\frac{{a – b}}{{a + b}}} \sqrt {\frac{{{a^2} + 2ab + {b^2}}}{{{a^2} – {b^2}}}}$$
A. $$a + b$$
B. $$\sqrt {\frac{{a + b}}{{a – b}}}$$
C. $$a – b$$
D. $$1$$

29. What is the simplified form of $$\sqrt{{\sqrt {343} }} \times \sqrt {28}$$
A. $$\sqrt {14}$$
B. 7
C. $$\sqrt 2$$
D. 14

30. The simplest form of $$\frac{{3\sqrt {24} – 2\sqrt {18} }}{{ – \sqrt 2 }}$$ is equal to:
A. $$6\left( {1 – \sqrt 3 } \right)$$
B. $$6\left( {\sqrt {3} – 1} \right)$$
C. $$6\sqrt 2 \left( {\sqrt 3 – 1} \right)$$
D. $$6\sqrt {2} \left( {1 – \sqrt 3 } \right)$$

31. What is the simplified form of  $$6\sqrt {2} \left( {1 – \sqrt 3 } \right)$$  for a$$a \ge 0?$$ ?
A. $$\sqrt{{{a^6}}}$$
B. $$\sqrt{{{a^2}}}$$
C. $$\sqrt{{{a^4}}}$$
D. a

32. The simplified form of $${\left( {\frac{{{{27}^{\frac{1}{4}}} \times {9^{\frac{{ – 1}}{6}}}}}{{{3^{\frac{1}{6}}}}}} \right)^{ – 4}}$$ is equal to:
A. 3
B. -3
C. $$\frac{1}{3}$$
D. $$\frac{{ – 1}}{3}$$

33. Which of the following is an irrational number?
A. $$\sqrt {\frac{{10}}{9} – \sqrt {\frac{1}{{81}}} }$$
B. $${\left( {\sqrt 5 – 1} \right)^2}$$
C.$$\sqrt{3}\left( {\sqrt{9} – \frac{1}{{\sqrt{3}}}} \right)$$
D. 6.$$\overline {68} – 3.\overline {45}$$

34. What is the simplified form of $$\frac{{\sqrt {3 + \sqrt 5 } }}{3}$$+$$\frac{{\sqrt {3 – \sqrt 5 } }}{3}$$
A. $$\frac{{\sqrt {30} }}{3}$$
B. $$\frac{{\sqrt {50} }}{3}$$
C. $$\frac{{\sqrt {10} }}{3}$$
D. 2

35. What are the values of x & y respectively, if $$\frac{{5 + 2\sqrt 3 }}{{7 + 4\sqrt 3 }}$$= x+y$$\sqrt 3$$ ?
A. -11, 6
B. 11, -6
C. 3, 4
D. -5, 7

36. If x=$$\frac{{3 + 2\sqrt 2 }}{{3 – 2\sqrt 2 }}$$, then$$x + \frac{1}{x}$$   is equal to:
A. 34
B. 37
C. $$9\sqrt 2$$
D. 9-2$$\sqrt 2$$

37. The simplified form of  $$\sqrt {16\sqrt{{{a^6}{b^{12}}}}}$$ is equal to:
A. $$.4{a^2}{b^4}$$
B. $$\left| a \right|{b^2}$$
C. 16$$\left| a \right|{b^2}$$
D. 4$$a{b^2}$$

38. Which of the following is not true?
A. $${2^{\sqrt 2 }} < {2^2}$$
B. $${(1/2)^{( – 3)}} < {(1/2)^{( – 4)}}$$
C. $${\left( {\frac{1}{2}} \right)^4} < {\left( {\frac{1}{2}} \right)^3}$$
D. $${2^\pi } = {2^{\frac{{22}}{7}}}$$

39. The simplified form of $$\sqrt {11 – 2\sqrt {30} }$$ is equal to:
A. $$\sqrt 6 + \sqrt 5$$
B. $$\sqrt 5 – \sqrt 6$$
C. $$\sqrt 6 – \sqrt 5$$
D. $$\sqrt 6 – \sqrt 2$$

40. Which of the following is not true?
A. Every element in Z has an additive inverse.
B. N is the not closed under the operation division.
C. 0 is identity element in the operation addition of real numbers
D. none

41. When the expression   $$\left( {\frac{{\frac{{\sqrt 3 – 1}}{{\sqrt 3 + 1{\rm{}}}} – \left( {\frac{{1 + \sqrt 3 }}{{\sqrt 3 – 1}}} \right)}}{{{{\left( {1 + \sqrt 2 } \right)}^2} – 3}}} \right)$$ is simplified the result is equal to:
A. $$\frac{{ – \sqrt 6 }}{2}$$
B.$$– \sqrt 6$$
C. $$– 2\sqrt 6$$
D. $$\frac{{\sqrt 6 }}{2}$$

42. $${\left( {{a^{ – 1}} + {b^{ – 1}}} \right)^{ – 1}}$$ is equal to__.
A. $$\frac{{a + b}}{{ab}}$$
B. $$\frac{{ab}}{{a + b}}$$
C. $$\frac{{a\left( {a + b} \right)}}{b}$$
D. $$\frac{{b\left( {a + b} \right)}}{a}$$

43. When the denominator of $$\frac{{\sqrt 5 – \sqrt 2 }}{{\sqrt 5 + \sqrt 2 }}$$ is rationalized it gives:
A. $$\frac{{3 – 2\sqrt {10} }}{3}$$
B. $$\frac{{7 – 2\sqrt {10} }}{3}$$
C. $$\frac{{3 + 2\sqrt {10} }}{3}$$
D. $$\frac{{3 + 2\sqrt {10} }}{3}$$

44. Which of the following is not true?
A. $$7\sqrt {3\,} \,\, – \,\,\,2\sqrt 3 \,\, = \,\,5\sqrt 3$$
B. $$\sqrt 3 \,\, \times \sqrt {27} \,\, = \,9$$
C. $$\sqrt {45} \,\, \div \sqrt 5 = 3$$
D. $$\sqrt {8\,} \,\, + \,\,\sqrt {32} \,\, = \,\,\sqrt {40}$$

45. If a =$$\frac{{\sqrt 3 – \sqrt 2 }}{{\sqrt 3 + \sqrt 2 }}$$  and ab =1, then the value of $${a^2} + 2ab + {b^2}is:$$ is
A. 100
B. $$\frac{{81}}{{41}}$$
C. 2
D. 3

46. The simplest form of $$\frac{{\left( {\frac{{\sqrt 2 – 1}}{{\sqrt 2 + 1}} – \left( {\frac{{\sqrt 2 + 1}}{{\sqrt 2 – 1}}} \right)} \right)}}{{4\sqrt {2} }}$$   is equal to:
A. 0
B. 1
C. -1
D. $$\frac{{ – 7}}{4}$$

47. What is the simplified form of  $$\sqrt{{\sqrt {343} }} \times \sqrt {28}$$  ?
A. $$\sqrt {14}$$
B. 7
C. 7$$\sqrt 2$$
D. 14

48. If   x =  $$\frac{{5 – 2\sqrt 3 }}{{5 + 2\sqrt 3 }}$$, then   x +$$\frac{1}{x}$$  is equal to:
A. $$\frac{{13}}{{17}}$$
B. $$\frac{6}{{17}}$$
C. $$\frac{{5\sqrt 3 }}{2}$$
D. $$\frac{{74}}{{13}}$$

49. When$$\sqrt {\frac{{3 + \sqrt 5 }}{2}}$$  +  $$\sqrt {\frac{{3 – \sqrt 5 }}{2}}$$  is simplified the result is equal to:
A. $$\sqrt 5$$
B. 2$$\sqrt {\frac{3}{2}}$$
C. 2
D. 2$$\sqrt {3 + \sqrt 5 }$$

50.  When expression $$\frac{{\frac{4}{{27}}\sqrt {0.0009} }}{{0.01\left( {2\frac{2}{{9}} – \frac{4}{3}} \right)}}$$  is simplified it gives:
A. 1
B. 2
C. $$\frac{1}{2}$$
D. $$\frac{1}{3}$$

51 .If x =$${\left( {{2^2}} \right)^3}$$  and y=$${2^{\left( {{2^3}} \right)}}$$, then which of the following is true?
A. $$x = y$$
B. $$X > Y$$
C.  $$x < y$$
D.  $$2x = y$$

52. When the expression $$\frac{{{{\left( {0.3} \right)}^2}\left( {1\frac{3}{4} + \frac{1}{3}} \right)}}{{{{\left( {0.1} \right)}^3}\left( {2\frac{1}{3} – \frac{1}{4}} \right)}}$$ is simplified, the result is equal to:
A. 0.
B. 90
C. 900
D. 0.9

53. If we simplify  $${\left( {\sqrt {3 \times {2^2} + \sqrt {{2^3} + \sqrt {{4^3}} } } } \right)^{\frac{{ – 1}}{2}}}$$  the result is:
A. $$\frac{{ – 1}}{2}$$
B. $$\frac{1}{{2\sqrt 3 }}$$
C. $$\frac{1}{2}$$
D. -$$2\sqrt 3$$

54. The simplification form of $$\frac{5}{{\sqrt 7 – \sqrt 2 }}$$ is
A. $$\sqrt {7} + \sqrt {2}$$
B. $$\sqrt 7 – \sqrt 2$$
C. 1
D. $$\sqrt 9$$

55.  What is the simplified form of the expression$$\frac{{\sqrt{{192}}}}{{\sqrt{6}}}$$?
A. $$\frac{{32}}{5}$$
B. $$\frac{1}{{32}}$$
C. 32
D. 2

56. Which of the following statements is necessary true for any real number a and a positive integer n?
A. $${a^n} + {a^n} = {a^{2n}}$$
B. $${a^0} = 1$$
C. $${a^{ – n}}$$=$${(\frac{1}{a})^{ – n}}$$
D. $${a^n}x{a^n} = {a^{2n}}$$

57. Which of the following pairs of prime numbers are twin primes?
A. 3and7
B. 5and7
C. 7and11
D. 11and17

58. If $$a = \sqrt 3 – 1$$and $$= \sqrt 3 + 1$$ , then which of the following is not true?
A. $$a + b = \sqrt 6$$
B. $$a – b = – 2$$
C. $$ab = 2$$
D. $$a + b = 2\sqrt 3$$

59.What is the simplified form of the expression $$\sqrt{{\frac{1}{7}}}.\sqrt{{\frac{1}{{27}}}}$$.$$\sqrt{{\frac{7}{{81}}}}?$$
A. $$\frac{1}{7}$$
B. $$\frac{1}{3}$$
C. $$\frac{1}{9}$$
D. $$\frac{1}{9}$$

60. When the denominator of $$\frac{{\sqrt 5 – \sqrt 2 }}{{\sqrt 5 + \sqrt 2 }}$$  is rationalized it gives:
A. $$\frac{{3 – 2\sqrt {10} }}{3}$$
B.  $$\frac{{7 – 2\sqrt {10} }}{3}$$
C. $$\frac{{3 + 2\sqrt {10} }}{3}$$
D. $$\frac{{7 + 2\sqrt {10} }}{3}$$

61. If m and n are positive integers, then which of the following is equal to $${n^{({m^2})}}?$$
A. $${n^m}{n^m}$$
B. nm + nm
C. (nm)2
D. (nm)m

62. To rationalize the denominator of  $$\frac{{5 + \sqrt 2 }}{{\sqrt{{{2^2}}}}}$$ which one of the following expression is used as rationalizing factor?
A. $$\frac{{\sqrt{{{2^2}}}}}{{\sqrt{{{2^2}}}}}$$
B. $$\frac{{\sqrt{{{2^5}}}}}{{\sqrt{{{2^5}}}}}$$
C. $$\frac{{\sqrt{{{2^4}}}}}{{\sqrt{{{2^4}}}}}$$
D. $$\frac{{\sqrt{{{2^9}}}}}{{\sqrt{{{2^9}}}}}$$

63. Which one of the following is true?
A. If a natural number is not prime, then it is composite.
B. If a natural number is even, then it is composite.
C. The sum of any two prime numbers is prime.
D. None of the above.

64. One of the following numbers is negative?
A. $$3\, – \,\sqrt 5$$
B. $$– \sqrt 3 \, + \,1$$
C. -$$\sqrt 7 \, + \,3$$
D. $$4\, – \,\sqrt {15}$$

65. Which of the following set is not closed under addition?
A. The set of natural numbers
B. The set of integers
C. The set of irrational numbers
D. The set of real numbers

66. Which one of the following is correct?
A. $$\mathop 3\nolimits^{\frac{1}{4}} \mathop {27}\nolimits^{\frac{1}{4}} \,\, = \,\,9$$
B. $$\sqrt(3]{5}\, \times \,\sqrt{5}\,\, = \,\,5$$
C. $$\frac{{\sqrt{{64}}}}{{\sqrt[{}]{2}}}\,\, = \,\,2$$
D. $$\sqrt{{\mathop {( – 5)}\nolimits^6 }}\,\,\, = \,\,\, – 5$$

67. Which of the following pairs of radicals are not like radicals?
A. $$\sqrt {48} \,\,\,\,\& \,\,\,\,\,\sqrt {27}$$
B. $$\sqrt {288} \,\,\,\,\& \,\,\,\,\,\sqrt {200}$$
C. $$\frac{1}{5}\sqrt {75} \,\,\,\,\& \,\,\,\,\, – 2\sqrt 3$$
D. $$\sqrt {300} \,\,\,\,\& \,\,\,\,\,\sqrt {50}$$

68. The LCM of two numbers is the product of the numbers when the numbers are:
A. composite
B. even
C. relatively prime
D. odd

69. The LCM of two numbers is 1440 and their GCF is 1, if one of the numbers is 32, then what is the other number?
A. 36
B. 45
C. 48
D. 38

70. For $$a \ge 0$$, p and q any positive integers. Which of the following is not equal to $${a^{\frac{p}{q}}}$$ ?
A. $$\sqrt[q]{{{a^p}}}$$
B. $${\left( {{a^{\frac{1}{q}}}} \right)^p}$$
C. $${\left( {\sqrt[q]{a}} \right)^p}$$
D. $$\sqrt[p]{{{a^q}}}$$

71. The simplified form of $$\frac{{\sqrt {144{x^5}{y^2}} }}{{2\sqrt {xy} }}$$ is equal to:
A. $$6x\sqrt y$$
B. $$6\sqrt {xy}$$
C. 6$${x^2}\sqrt y$$
D. $$\sqrt {6xy}$$

72. For any two positive real numbers a and b and for all integers n$$\ge$$2, which of the following is not true?
A. $${a^{\frac{1}{n}}}$$$${b^{\frac{1}{n}}}$$  =$${\left( {ab} \right)^{\frac{1}{n}}}$$
B. $$\frac{{{a^{\frac{1}{n}}}}}{{{b^{\frac{{1}}{n}}}}}$$ =  $${\left( {\frac{a}{b}} \right)^{\frac{1}{n}}}$$
C. $$\sqrt[n]{{{a^n}}}$$$$\sqrt[n]{b}$$  =  $$\sqrt[n]{{ab}}$$
D. $$\sqrt[n]{a}$$+ $$\sqrt[n]{b}$$ =$$\sqrt[n]{{a + b}}$$

73. Which of the following is prime factorization of 180?
A. $${2^2}{.3^2}.5$$
B. $${3^2}{.5^2}$$
C. $${2.3^2}.5$$
D. $${2^2}.3.5$$

74. The simplified form of the expression $$\sqrt{{\sqrt {216} }}$$  is:
A. $$6{^{\frac{1}{{2}}}}$$
B. $$6{^{\frac{1}{{5}}}}$$
C. $$6{^{\frac{1}{{3}}}}$$
D. $$216{^{\frac{5}{{6}}}}$$

75. Which of the following sets is closed under the operation multiplication of real numbers?
A. $$\left\{ { – 4,0,4} \right\}$$
B. $$\left\{ { – 1,0,1} \right\}$$
C. $$\left\{ { – 2,0,2} \right\}$$
D. none

 No Explanations 1. The answer is B.b/c 151 has no more than two factors 2. The answer is C:  $$1008 = 24x32x7$$, therefore the answer is C 3. The answer is D. since$$2x3x5 = 30$$. The factors of $$30 = \left\{ {1,2,3,5,6,10,15,30} 4. The answer is A:Let \(d = 5.575575 \ldots ..\left( 1 \right)$$                1000d=5575.575575….(2)       subtract (1) from (2)$$999d = 5570$$$$d = \frac{{5570}}{{999}}$$ 5. The answer is A: The number 3.765765…. is equal to:Let $$d = 3.765765 \ldots$$         (1) by multiplying both sides by 1000 we get (2)     1000d =3765.765765…             (2)                      $$999d = 3762$$  subtract (1) from (2) $$d = \frac{{3762}}{{999}} = \frac{{418}}{{111}}$$ 6. The answer is A : $$1800 = {2^3}{.3^2}{.5^2},756 = {2^2}{.3^3}.7$$.$$GCF\left( {1800,756} \right) = {2^2}{.3^2} = 36andLCM\left( {1800,756} \right) = {2^3}{.3^3}{.5^2}.7 = 37800$$ 7. The answer is A. b/c 126 is divisible bye 2 and 3 8. The Answer is B: $$\sqrt{{\frac{1}{5}}}\sqrt{{\frac{1}{9}}}\sqrt{{\frac{5}{{27}}}} = \sqrt{{\frac{1}{5}.\frac{1}{9}.\frac{5}{{27}}}} = \sqrt{{\frac{1}{{{3^5}}}}} = \frac{1}{3}$$, 9. The answer is C : $$\Rightarrow 5 < \sqrt {35} < 6$$$$\Rightarrow 5 – 6 < \sqrt {35} – 6 < 6 – 6$$$$\Rightarrow – 1 < \sqrt {35} – 6 < 0$$ Therefore it is located between -1 and 0. 10. The answer is B: (apply the same way of question 9 above.) 11. The answer is D:  40=23x5, 1400=23x52x7,  The number should be 175 since the required number is odd 12. The answer is C. 13. The answer is D:  $$GCF\left( {a,b} \right)xLCM\left( {a,b} \right) = axb \Rightarrow 12×180 = 36xb \Rightarrow b = 60$$. 14. The answer is C :  $$\frac{{\sqrt {5 – 2\sqrt 6 } .\sqrt {5 + 2\sqrt 6 } }}{{\sqrt{{ – 8}}}} = \frac{{\sqrt {\left( {5 – 2\sqrt 6 } \right)\left( {5 + 2\sqrt 6 } \right)} }}{{\sqrt{{ – 8}}}} = \frac{{\sqrt {25 – 24} }}{{ – 2}} = \frac{{ – 1}}{2}$$, 15. The answer is D, b/c the average of two numbers is always located between them 16. The answer is C 17. The answer is EA. $$\frac{{\sqrt {12} }}{{\sqrt 3 }} = \frac{{\sqrt 4 \sqrt 3 }}{{\sqrt 3 }} = 2 \leftarrow rational$$B. $$\frac{{\sqrt {{4^{\frac{1}{2}}}} }}{{\sqrt 2 }} = 1 \leftarrow rational$$C. $$\frac{{\sqrt 2 – 1}}{{1 – \sqrt 2 }} = \frac{{\sqrt 2 – 1}}{{1 – \sqrt 2 }}\frac{{\sqrt 2 + 1}}{{1 + \sqrt 2 }} = \frac{{2 – 1}}{{1 – 2}} = – 1 \leftarrow rational$$D. 7.545454…E. None 18 The answer is B 19 The answer is B 20 The answer is  C:  B/c the sum of rational and irrational numbers is always irrational 21 The answer is B 22 The answer is D 23 The answer is A. 24 The answer is B 25 The answer is B:  0.001050 has 4 significant figures since zeros before first non-zero are insignificant 26 The answer is C 27. B, $$\sqrt {64\sqrt{{{y^{12}}{z^6}}}} =8\sqrt {{y^4}{z^2}} = 8{y^2}\left| z \right|$$ 28. The answer is D: $$\sqrt {\frac{{a – b}}{{a + b}}} .\sqrt {\frac{{{a^2} + 2ab + {b^2}}}{{{a^2} – {b^2}}} = } \sqrt {\frac{{a – b}}{{a + b}}} .\sqrt {\frac{{{{\left( {a + b} \right)}^2}}}{{\left( {a – b} \right)\left( {a + b} \right)}}} = 1$$, 29. The answer is D:  $$\sqrt{{\sqrt {343} }} \times \sqrt {28}$$ =$$\sqrt{{\sqrt {{7^3}} }}\sqrt {4.7} = \sqrt 7 \left( {2\sqrt 7 } \right)$$= $$2×7 = 14$$, 30. The answer is A: $$\frac{{3\sqrt {24} – 2\sqrt {18} }}{{ – \sqrt 2 }} = 3\sqrt {4×6} – \frac{{3\sqrt {4×6} – 2\sqrt {9×2} }}{{ – \sqrt 2 }} = \frac{{6\sqrt 6 – 6\sqrt 2 }}{{ – \sqrt 2 }} = – 6\sqrt 3 + 6$$, 31. The answer is B:  $$\sqrt{{{a^2}.\sqrt{{{a^2}}}}} = \sqrt{{{a^2}{a^{\frac{2}{3}}}}} = \sqrt{{{a^{\frac{8}{3}}}}} = {\left( {{a^{\frac{8}{3}}}} \right)^{\frac{1}{4}}} = {a^{\frac{2}{3}}} = \sqrt{{{a^2}}}$$, 32. The answer is C:  $${\left( {\frac{{{{27}^{\frac{1}{4}}} \times {9^{\frac{{ – 1}}{6}}}}}{{{3^{\frac{1}{6}}}}}}\right)^{ – 4}} = {\left( {\frac{{{{\left( {{3^3}} \right)}^{\frac{1}{4}}}{{\left( {{3^2}}\right)}^{\frac{{ – 1}}{6}}}}}{{{3^{\frac{1}{6}}}}}} \right)^{ – 4}} = {\left( 3 \right)^{ – 1}}= \frac{1}{3}$$ 33. Which of the following is an irrational number?A. $$\sqrt {\frac{{10}}{9} – \sqrt {\frac{1}{{81}}} } = \sqrt {\frac{{10}}{9} – \frac{1}{9}} = 1 \leftarrow$$ rationalB. $${\left( {\sqrt 5 – 1} \right)^2} = 6 – 2\sqrt 5 \leftarrow$$irrational.C.  $$\sqrt{3}\left( {\sqrt{9} – \frac{1}{{\sqrt{3}}}} \right) \leftarrow rational$$ D. 6.3$$\overline {68} – 3.\overline {45} \leftarrow rational$$ 34 The correct answer is C:Let  $$x = \frac{{\sqrt {3 + \sqrt 5 } }}{3}$$+$$\frac{{\sqrt {3 – \sqrt 5 } }}{3}$$$${x^2} = {\left( {\frac{{\sqrt {3 + \sqrt 5 } }}{3}{\rm{}} + \frac{{\sqrt {3 – \sqrt 5 } }}{3}} \right)^2}$$$${x^2} = \frac{{3 + \sqrt 5 + 2\sqrt {\left( {3 – \sqrt 5 } \right)\left( {3 + \sqrt 5 } \right)} + 3 – \sqrt 5 }}{9}$$$$9{x^2} = 6 + 2\sqrt {9 – 5}$$$$9{x^2} = 10$$$$x = \frac{{\sqrt {10} }}{3}$$Therefore, $$\frac{{\sqrt {3 + \sqrt 5 } }}{3}$$+$$\frac{{\sqrt {3 – \sqrt 5 } }}{3} = \frac{{\sqrt {10} }}{3}$$ 35. The correct answer is B: $$\frac{{5 + 2\sqrt 3 }}{{7 + 4\sqrt 3 }}$$= x+y$$\sqrt 3$$, $$\frac{{5 + 2\sqrt 3 }}{{7 + 4\sqrt 3 }}$$=11-6√3⇒ $$x = 11andy = – 6$$ 36. The answer is A. 37 The answer is B : $$\sqrt {16\sqrt{{{a^6}{b^{12}}}}} = 4\sqrt {{a^2}{b^4}} = 4\left| a \right|{b^2}$$. 38. The answer is D. 39. The answer is C. 40 The answer is D. 41. The answer is B. 42. The answer is B. 43. The answer is B. 44. The answer is D. b/c unlike radicals cannot be added as a single radical 45. The correct answer is A 46. The answer is C : $$\frac{{\left( {\frac{{\sqrt 2 – 1}}{{\sqrt 2 + 1}} – \left( {\frac{{\sqrt 2 + 1}}{{\sqrt 2 – 1}}} \right)} \right)}}{{4\sqrt {2} }} = \frac{{\frac{{{{\left( {\sqrt 2 – 1} \right)}^2} – {{\left( {\sqrt 2 + 1} \right)}^2}}}{{\left( {\sqrt 2 + 1} \right)\left( {\sqrt 2 – 1} \right)}}}}{{4\sqrt 2 }} = \frac{{\frac{{ – 4\sqrt 2 }}{1}}}{{4\sqrt 2 }} = – 1$$ 47. The Answer is D: $$\sqrt{{\sqrt {343} }} \times \sqrt {28} = \sqrt{{\sqrt {{7^3}} }}.\sqrt {4.7} = \sqrt 7 .2\sqrt 7 = 14$$ 48. The answer is D : x +$$\frac{1}{x} =$$$$\frac{{5 – 2\sqrt 3 }}{{5 + 2\sqrt 3 }} + \frac{{5 + 2\sqrt 3 }}{{5 – 2\sqrt 3 }} = \frac{{74}}{{13}}$$. 49. The answer is A:Let $$x = \sqrt {\frac{{3 + \sqrt 5 }}{2}}$$  +  $$\sqrt {\frac{{3 – \sqrt 5 }}{2}}$$ $${x^2} = {\left( {\sqrt {\frac{{3 + \sqrt 5 }}{2}} {\rm{}} + {\rm{}}\sqrt {\frac{{3 – \sqrt 5 }}{2}} } \right)^2}$$$$2{x^2} = 6 + 2\sqrt 4$$$$x = \sqrt 5$$ 50. The answer is C:$$\frac{{\frac{4}{{27}}\sqrt {0.0009} }}{{0.01\left( {2\frac{2}{{9}} – \frac{4}{3}} \right)}} = \frac{{\frac{4}{{27}}\sqrt {\frac{9}{{10000}}} }}{{\frac{1}{{100}}\left( {\frac{{20}}{9} – \frac{4}{3}} \right)}} = \frac{{\frac{4}{{27}}\frac{3}{{100}}}}{{\frac{1}{{100}}\frac{8}{9}}} = \frac{1}{2}$$ 51 If x =$${\left( {{2^2}} \right)^3} = {2^6} = {2^8} \Rightarrow 6 > 8$$ 52. The answer is B:$$\frac{{{{\left( {0.3} \right)}^2}\left( {1\frac{3}{4} + \frac{1}{3}} \right)}}{{{{\left( {0.1} \right)}^3}\left( {2\frac{1}{3} – \frac{1}{4}} \right)}} = \frac{{0.09\left( {\frac{{25}}{{12}}} \right)}}{{0.001\left( {\frac{{25}}{{12}}} \right)}} = 90$$ 53 The answer is C:$${\left( {\sqrt {3 \times {2^2} + \sqrt {{2^3} + \sqrt {{4^3}} } } } \right)^{\frac{{ – 1}}{2}}}$$$${\left( {\sqrt {12 + \sqrt {8 + \sqrt {64} } } } \right)^{\frac{{ – 1}}{2}}}$$$${\left( {\sqrt {12 + \sqrt {8 + \sqrt {64} } } } \right)^{\frac{{ – 1}}{2}}}$$$$= {\left( {\sqrt {12 + \sqrt {16} } } \right)^{\frac{{ – 1}}{2}}}$$$$= {\left( {\sqrt {12 + \sqrt {16} } } \right)^{\frac{{ – 1}}{2}}}$$$$= {\left( {\sqrt {12 + \sqrt {16} } } \right)^{\frac{{ – 1}}{2}}}$$$$= {\left( 4 \right)^{ – \frac{1}{2}}}$$$$= \frac{1}{{\sqrt 4 }}$$$$= \frac{1}{2}$$ 54 The Answer is A :$$\frac{5}{{\sqrt 7 – \sqrt 2 }} = \frac{5}{{\sqrt 7 – \sqrt 2 }}.\frac{{\sqrt 7 + \sqrt 2 }}{{\sqrt 7 + \sqrt 2 }} = \frac{{5\left( {\sqrt 7 + \sqrt 2 } \right)}}{5} = \sqrt 7 + \sqrt 2$$ 55. The answer is D :$$\frac{{\sqrt{{192}}}}{{\sqrt{6}}} = \sqrt{{\frac{{192}}{6}}} = \sqrt{{32}} = 2$$ 56. The answer is A 57 The answer is B. 58 The answer is B. 59 The answer is A: $$\sqrt{{\frac{1}{7}}}.\sqrt{{\frac{1}{{27}}}}$$.$$\sqrt{{\frac{7}{{81}}}} = \sqrt{{\frac{1}{7}.\frac{1}{{27}}.\frac{7}{{81}}}} = \sqrt{{\frac{1}{{{3^7}}}}} = \frac{1}{3}$$ 60 The answer is D 61 The answer is D: $${n^{{m^2}}} = {\left( {{m^m}} \right)^m}$$, the answer is D. 62. The answer is B:       The rationalizing factor of  $$\frac{{5 + \sqrt 2 }}{{\sqrt{{{2^2}}}}}$$  is  $$\frac{{\sqrt{{{2^5}}}}}{{\sqrt{{{2^5}}}}}$$ 63 The answer is D:Justifications:A. If a natural number is not prime, then it is composite. This statement is false since 1 is not prime and it is not composite.                                B. If a natural number is even, then it is composite. Incorrect! b/c 2 is even but it is not composite.C. The sum of any two prime numbers is prime. Incorrect! b/c sum of two prime numbers may not be prime. Eg. 3+7=10.D. None of the above 64 The answer is B 65. The answer is C: The set of irrational numbers is not closed under the operation addition. Since the sum of any two irrational numbers may not be irrational. 66 The answer is C. 67 The answer is A. 68 The answer is C 69 The answer is B 70 The answer is D:  b/c $${a^{\frac{p}{q}}} \ne \sqrt[p]{{{a^q}}}$$ 71 The answer is A : $$\frac{{\sqrt {144{x^5}{y^2}} }}{{2\sqrt {xy} }} = \frac{{6\sqrt {{x^5}{y^2}} }}{{\sqrt {xy} }} = 6{x^2}\sqrt y$$, 72 The answer is D: B/c  +  $$\sqrt[n]{a} + \sqrt[n]{b} \ne \sqrt[n]{{a + b}}$$ 73 The correct answer is A 74 The answer is A. $$\sqrt{{\sqrt {216} }} = \sqrt{{\sqrt {{6^3}} }} = \sqrt 6 = {6^{\frac{1}{2}}}$$ 75 The answer is B :{-1, 0, 1}is closed under the operation multiplication