Simplification and Approximate value

BODMAS – ( bracket of division multiplication addition and subtraction)

Step 1 –  First of all solve brackets

Step 2 –  After that solve powers and surds ( squares , cubes , square root , cube root etc.)

Step 3 – After that solve multiply or divide from left to right which comes first .

Step 4 – Finally solve addition or subtraction from left to right which comes first.

 

Laws of integral exponents

1. [latex]a^{x}[/latex] = [latex]a\times a\times a[/latex] ……………. [latex]a\times a[/latex]   (x times)

2. [latex]a^{x}\times a^{y}[/latex] = [latex]a^{x + y}[/latex]

3. [latex]\frac{a^{x}}{a^{y}}[/latex] = [latex]a^{x – y}[/latex]

4. [latex](a^{x})^{y}[/latex] = [latex]a^{xy}[/latex]

5. [latex]\sqrt[n]{a}[/latex] = [latex]a^{\frac{1}{n}}[/latex]

6. [latex]\sqrt{a}[/latex] = [latex]a^{\frac{1}{2}}[/latex]

7.  [latex](a^{x})^{y}[/latex]  [latex]\neq[/latex] [latex]a^{x^{y}}[/latex]

Example (7) [latex](2^{2})^{3}[/latex] = [latex]2^{6}[/latex]  = 64

[latex]2^{2^{3}}[/latex] = [latex]2^{8}[/latex] = 256

64 [latex]\neq[/latex] 256

8. [latex](\sqrt[m]{(\sqrt[n]{x})^{a}})^{b}[/latex] =[latex]x^{\frac{a\times b}{m\times n}}[/latex]

 

Short tricks to find square roots

First remember the squares from 1 to 9

[latex]1^{2}[/latex] = 1

[latex]2^{2}[/latex] = 4

[latex]3^{2}[/latex]= 9

[latex]4^{2}[/latex]= 16

[latex]5^{2}[/latex]= 25

[latex]6^{2}[/latex] = 36

[latex]7^{2}[/latex] = 49

[latex]8^{2}[/latex] = 64

[latex]9^{2}[/latex] = 81

 

1. [latex]\sqrt{2809}[/latex] =  ?                                                        

First check the unit place of this number (which is 9)

3^2 = 9

And 7^2 = 49

The unit place of the number is 3 or 7

Ignore the last two digits (09)

Remaining digits = 28

Now 28 is in between 5^2 and 6^2 ( take the smaller one i.e. 5 for the square root except unit place which is 3 or 7 )

5*6 = 30 > 28

Since 28 is small than 30 so we take smaller one in 3 and 7

[latex]\sqrt{2809}[/latex] =  53

 

2. [latex]\sqrt{5776}[/latex] = ?

57 and 76

4^2 = 16   and  6^2 = 36

7^2 < 57 < 8^2

7*8 = 56 < 57

Since 57 is greater than 56, so we take larger in 4 and 6

[latex]\sqrt{5776}[/latex] = 76

 

3.  [latex]\sqrt{12544}[/latex] = ?

125 and 44

2^2 = 4 and 8^2 = 64

125 lies between the squares of 11 and 12 (smaller one is 11)

11*12 = 132 > 125

Since 125 is smaller than 132, so we take smaller in 2 and 8

[latex]\sqrt{12544}[/latex] = 112

 

Short tricks to find cube roots

First remember the cubes from 1 to 9

[latex]1^{3}[/latex] = 1

[latex]2^{3}[/latex] = 8

[latex]3^{3}[/latex] = 27

[latex]4^{3}[/latex] = 64

[latex]5^{3}[/latex] = 125

[latex]6^{3}[/latex] = 216

[latex]7^{3}[/latex] = 343

[latex]8^{3}[/latex] = 512

[latex]9^{3}[/latex] = 729

 

1. [latex]\sqrt[3]{1744}[/latex] 

First check the unit place of the given number (which is 4)

Now find from the above cubes that which cubes has unit place 4 (which is 4)

So unit digit of our cube root is 4

Now ignore the last three digits 744

Remaining digits = 2

Now 1^3< 2 < 2^3 ( take smaller one which is 1)

So our cube root is 14

 

2. [latex]\sqrt[3]{17576}[/latex] 

Unit place of this number = 6

So unit place of our cube root is 6 (6^3= 216)

Ignore the last three digits

Remaining digits = 17

2^3< 17 < 3^3 ( take smaller one which is 2 )

So our cube root is 26

 

3.  [latex]\sqrt[3]{3778688}[/latex]                                             

Unit place of this number = 8

So unit place of our cube root is 2 (2^3= 8)

Ignore the last three digits

Remaining digits = 778

9^3< 778 < 10^3 ( take smaller one i.e. 9)

Our cube root is 92

 

Few questions of simplification and approximate value

1. What come in place of ? [latex][(7 – 5) + 2^{3} – 4]\div2 – 5 = ?[/latex]

a) 3                          b) 2                         c) -3                       d) -2

Answer – d) -2

Solution [latex][(7 – 5) + 2^{3} – 4]\div2 – 5[/latex] = [latex][2 + 2^{3} – 4]\div2 – 5[/latex]

= [latex][2 + 8 – 4]\div2 – 5[/latex]

=  [latex][6]\div2 – 5[/latex]

= 3 – 5   = -2

 

2. [latex]\sqrt{\sqrt{{14641}}}[/latex] is equal to

a) 10                         b) 11                            c) 21                            d) 9

Answer – b) 11

Solution [latex]\sqrt{\sqrt{11\times11\times11\times11}}[/latex]

= [latex]\sqrt{11\times11}[/latex] = 11

 

3. [latex]\sqrt{56+\sqrt{56+\sqrt{56+}}}[/latex]………………… is equal to

a) 8                         b) -7                          c) 6                             d) -14

Answer – a) 8

Solution : Let x = [latex]\sqrt{56+\sqrt{56+\sqrt{56+}}}[/latex] …………………….

squaring both sides

[latex]x^{2}[/latex] = 56 + [latex]\sqrt{56+\sqrt{56+\sqrt{56+}}}[/latex]……………………….

[latex]x^{2}[/latex] = 56 + x

[latex]x^{2}[/latex] – x  – 56 = 0

(x – 8) (x + 7) = 0

x  = 8  or x = -7

hence x = 8 ( This is because square root only gives positive value )

 

4. simplify  [latex]\frac{1}{8} + \frac{1}{120}+\frac{1}{330}+ \frac{1}{638}+\frac{1}{1044}+\frac{1}{1548}[/latex]

a) 6/43                     b) 13/54263                      c) 1/43                              d) 42/43

Answer – a) 6/43

 

Solution [latex]\frac{1}{8} + \frac{1}{120}+\frac{1}{330} + \frac{1}{638}+\frac{1}{1044}+\frac{1}{1548}[/latex]

= [latex]\frac{7}{7}[/latex] [ [latex]\frac{1}{8} + \frac{1}{120}+\frac{1}{330} + \frac{1}{638}+\frac{1}{1044}+\frac{1}{1548}[/latex] ]

= [latex]\frac{1}{7}[/latex] [ [latex]\frac{7}{8} + \frac{7}{120}+\frac{7}{330} + \frac{7}{638}+\frac{7}{1044}+\frac{7}{1548}[/latex] ]

= [latex]\frac{1}{7}[\frac{7}{1\times8}+\frac{7}{8\times15}+\frac{7}{15\times22}+\frac{7}{22\times29}+\frac{7}{29\times36}+\frac{7}{36\times43}][/latex]

= [latex]\frac{1}{7}[\frac{1}{1}-\frac{1}{8}+\frac{1}{8}-\frac{1}{15}+\frac{1}{15}-\frac{1}{22}+\frac{1}{22}-\frac{1}{29}+\frac{1}{29}-\frac{1}{36}+\frac{1}{36}-\frac{1}{43}+\frac{1}{43}][/latex]

=[latex]\frac{1}{7}[\frac{1}{1}-\frac{1}{43}][/latex]

= 6/43

 

5. Find the value of [latex]\sqrt{-3.5\times\sqrt{-3.5\times\sqrt{-3.5}}}[/latex]……………..

a) 3.5                             b) – 3.5                              c) 0                             d) 1

Answer – b) – 3.5

Solution Let x = [latex]\sqrt{-3.5\times\sqrt{-3.5\times\sqrt{-3.5}}}[/latex]

[latex]x^{2}[/latex] = -3.5 [latex]\sqrt{-3.5\times\sqrt{-3.5\times\sqrt{-3.5}}}[/latex]

[latex]x^{2}[/latex] = – 3.5 x

x = – 3.5

 

6. Simplify [latex]2+\frac{3}{5+\frac{6}{7+\frac{2}{3}}}[/latex]

a) 234/17                 b) 451 / 123                    c) 546/119                       d) 335/133

Answer – d) 335/133

Solution [latex]2+\frac{3}{5+\frac{6}{7+\frac{2}{3}}}[/latex]

= [latex]2+\frac{3}{5+\frac{6}{\frac{23}{3}}}[/latex]

= [latex]2+\frac{3}{5+\frac{18}{23}}[/latex]

= [latex]2+\frac{3}{\frac{133}{23}}[/latex]

= [latex]2+\frac{69}{133}[/latex]

=[latex]\frac{335}{133}[/latex]

 

7. If [latex]\frac{4}{\sqrt[3]{121}+1+\sqrt[3]{11}}[/latex] = A[latex]\sqrt[3]{11}+B\sqrt[3]{121}[/latex] + C, then A + B + C is equal to

a) 2/11                     b) 2/3                               c) 4/11                         d) 1/6

Answer – b) 2/3

solution  [latex]\frac{4}{\sqrt[3]{121}+1+\sqrt[3]{11}}[/latex] =  [latex]\frac{4}{\sqrt[3]{121}+1+\sqrt[3]{11}}[/latex] [latex]\times[/latex] [latex]\frac{\sqrt[3]{11}+1}{\sqrt[3]{11}+1}[/latex]

= [latex]\frac{\sqrt[3]{11}}{(11^{\frac{2}{3}} + 1 + 11^{\frac{1}{3}})(11^{\frac{1}{3}}+1)}[/latex]

= [latex]\frac{4}{12}\times(\sqrt[3]{11}+1)[/latex]

= [latex]\frac{1}{3}\sqrt[3]{11}+\frac{1}{3}[/latex]

A = 1/3 , B = 0 and C = 1/3

A + B + C = 1/3 + 1/3 = 2/3

 

8.  [latex](\sqrt[3]{(\sqrt[4]{13})^{6}})^{4}[/latex] is equal to

a) 13                       b) 2197                             c) 169                           d) 1

Answer – c) 169

Solution [latex](\sqrt[3]{(\sqrt[4]{13})^{6}})^{4}[/latex] = [latex](13)^{\frac{6\times4}{4\times3}}[/latex]

= [latex]13^{2}[/latex]  = 169

 

9. [latex]\frac{\sqrt{205209}}{\sqrt[3]{3442951}}[/latex] is equal to

a) 1                   b) 2                            c) 3                              d) 4

Answer – c) 3

Solution [latex]\sqrt{205209}[/latex] = 453

[latex]\sqrt[3]{3442951}[/latex] = 151

[latex]\frac{\sqrt{205209}}{\sqrt[3]{3442951}}[/latex] = 453/151 = 3

 

Problems for Practice

Directions (1-10): What value will come in place of the question mark (?) in each of the following equation ?
1. [latex]\frac{?}{529}=\frac{324}{?}[/latex]
a) 404
b) 408
c) 410
d) 414
e) 416

2. {2.002 + 7.9(2.8 – 1.4)} = ?
a) 11.312
b) 12.204
c)13.062
d) 14.442
e)11.006

3. [latex]\frac{18\times15-20}{(3.2+9.4)-7.6}=?[/latex]
a) 25
b) 100
c) 150
d) 50
e) none of these

4. [latex]\frac{(5.7\times5.7\times5.7+2.3\times2.3\times2.3)}{(5.7\times5.7+2.3\times2.3-5.7\times2.3)}[/latex] = ?
a) 2.3
b) 3.4
c) 5.7
d) 8.0
e) 1

5. [latex]456\div24\times38-958+364[/latex] = ?
a) 140
b) 128
c) 132
d) 136
e) 144

6. [latex]\sqrt{13.3225}[/latex] = ?
a) 3.45
b) 3.55
c) 3.65
d) 3.75
e) 3.85

7. [latex]3\frac{1}{4}+2\frac{1}{2}-1\frac{5}{6}=\frac{?^{2}}{10}+1\frac{5}{12}[/latex]
a) 25
b) [latex]\sqrt{5}[/latex]
c) 625
d) 15
e) 5

8. [latex](\sqrt{8}\times\sqrt{8})^{2}+9^{\frac{1}{2}}=?^{3}+\sqrt{8}-340[/latex]
a) 7
b) 19
c) 18
d) 9
e) none of these

9. [latex]\sqrt{?}=(78\times148)\div481[/latex]
a) 484
b) 529
c) 576
d) 625
e) 676

10. [latex]6\frac{2}{5}\times5\frac{5}{8}\times11\frac{11}{14}\div6\frac{2}{7}[/latex] = ?
a) 63.5
b) 64.5
c) 65.5
d) 66.5
e) 67.5

Directions (11-20): What approximate value will come in place of the question mark (?) in each of the following equation ?
11. 185% of 1359 + 18.5% of 1319 = ?
a) 2510
b) 2630
c) 2760
d) 2890
e) 3025

12. [latex]43.03\times27.96+11.98\times\sqrt[3]{42870}=?[/latex]
a) 1625
b) 1705
c) 1775
d) 1815
e) 1855

13. [latex]{(8.66^{2})\times13.98}\div\sqrt{50}[/latex]
a) 120
b) 130
c) 140
d) 150
e) 160

14. [latex]\sqrt{5475}\div4.98[/latex] = ?
a) 11
b) 15
c) 20
d) 24
e) 27

15. [latex]118.07\times1349[/latex] + 169% of 784 = ?
a) 2520
b) 2610
c) 2750
d) 2870
e) 2930

16. [latex]\sqrt{2300}\times\sqrt{240}[/latex] = ?
a) 685
b) 705
c) 815
e) 745
e) 635

17. [latex]119.00.\times14.987+21.04\times13.96[/latex] = ?
a) 2080
b) 2120
c) 2150
d) 2175
e) 2250

18. [latex](\frac{24}{9})^{2}\times\frac{399}{99}\div\frac{41}{899}[/latex] = ?
a) 600
b) 650
c) 700
d) 630
e) 750

19. [latex]-(4.99)^{3}+(29.98)^{2}-(3.01)^{4}[/latex] = ?
a) 550
b) 590
c) 620
d) 650
e) 690

20. [latex]44.4\times4.44\div7.98+\sqrt{2400}[/latex] = ?
a) 50
b) 60
c) 70
d)80
e) 90

Directions (1-10): What value will come in place of the question mark (?) in each of the following equation ?
21. [latex](0.2)^{\frac{3}{2}}\times0.008\div\frac{1}{\sqrt{0.2}}=(0.2)^{?}[/latex]
a) 1
b) 2
c) 3
d) 4
e) 5

22. [latex](7776)^{1.3}\times(36)^{1.25}\div(216)^{2}\div(1296)^{-1}=6^{?}[/latex]
a) 3
b) 4
c) 5
d) 6
e) 7

23. [latex]3\frac{2}{7}[/latex] of [latex]4\frac{5}{11}[/latex] of [latex]\frac{3}{35}[/latex] of 3080 = ?
a) 3864
b) 3948
c) 4014
d) 4124
e) none of these

24. [latex][\sqrt[5]{\sqrt[2]{38416}}]^{\frac{5}{2}}[/latex] = ?
a) 14
b) 196
c) 2744
d) 38416
e) none of these

25. [latex]\frac{3.673^{3}+7.327^{3}}{3.673^{2}+7.327^{2}-(3.673\times7.327)}[/latex] = ?
a) 10
b) 11
c) 12
d) 9
e) 13

Directions (26-30): What approximate value will come in place of the question mark (?) in each of the following equation ?
26. [latex]\sqrt{\sqrt{29585}+\sqrt{23100}}[/latex] = ?
a) 18
b) 20
c) 16
d) 22
e) 24

27. 48.5% of 7842 + ? % of 1318 = 4515
a) 42
b) 48
c) 54
d) 57
e) 60

28. [latex]\sqrt[3]{226980}[/latex] = ?
a) 59
b) 61
c) 63
d) 65
e) 67

29. [latex]\sqrt[3]{29790}\times\sqrt{1760}[/latex] = ?
a) 1200
b) 1250
c) 1300
d) 1350
e) 1400

30. [latex]1873\div84.05+40.81\times16.96[/latex] = ?
a) 700
b) 720
c) 740
d) 760
e) 780

 

Answers

1. d                                   2. c                                    3. d                                   4. d                                   5. c

6. c                                  7. e                                    8. a                                      9. c                                 10. e

11. c                                12. a                                   13. d                                  14. b                                 15. e

16. d                               17. a                                    18. d                                  19. e                                 20. c

21. d                              22. c                                 23. a                                   24. a                                   25. b

26. a                            27. c                                    28. b                                  29. c                                   30. b