Class 7 - Maths - Exponents and Powers

Exercise 13.1

Question 1:

Find the value of:

(i) 26            (ii) 93              (iii) 112               (iv) 54

Answer:

(i) 26 = 2 * 2 * 2 * 2 * 2 * 2 = 64

(ii) 93 = 9 * 9 * 9 = 729

(iii) 112 = 11 * 11 = 121

(iv) 54 = 5 * 5 * 5 * 5 = 625

 

Question 2:

Express the following in exponential form:

(i) 6 * 6 * 6 * 6 (ii) t * t  (iii) b * b * b * b (iv) 5 * 5 * 7 * 7 * 7 (v) 2 * 2 * a * a

(vi) a * a * a * c * c * c * c  * d

Answer:

(i) 6 * 6 * 6 * 6 = 64

(ii) t * t = t2

(iii) b * b * b * b = b4

(iv) 5 * 5 * 7 * 7 * 7 = 52 * 73

(v) 2 * 2 * a * a = 22 * a2

(vi) a * a * a * c * c * c * c  * d = a3 * c4 * d

 

Question 3:

Express each of the following numbers using exponential notations:

(i) 512          (ii) 343            (iii) 729             (iv) 3125

Answer:

(i) 512 = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 29

    Class_7_Exponents_&_Powers_Primefactorization_of_512                                                                               

(ii) 343 = 7 * 7 * 7 = 73

    Class_7_Exponents_&_Powers_Primefactorization_of_343                                                                               

(iii) 729 = 3 * 3 * 3 * 3 * 3 * 3 = 36

   Class_7_Exponents_&_Powers_Primefactorization_of_729                                                                                

(iv) 3125 = 5 * 5 * 5 * 5 * 5 = 55

  Class_7_Exponents_&_Powers_Primefactorization_of_3125                                                                                 

Question 4:

Identify the greater number, wherever possible, in each of the following:

(i) 43 and 34       (ii) 53 or 35    (iii) 28 or 82     (iv) 1002 or 2100       (v) 210 or 102

 Answer:

(i) 43 = 4 * 4 * 4 = 64

    34 = 3 * 3 * 3 * 3 = 81 

Since 64 < 81

So, 34 is greater than 43

(ii) 53 = 5 * 5 * 5 = 125

     35 = 3 * 3 * 3 * 3 * 3 = 243

Since 125 < 243

So, 35 is greater than 53  

(iii) 28 = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 256

       82 = 8 * 8 = 64

Since, 256 > 64

Thus, 28 is greater than 82

(iv) 1002 = 100 * 100 = 10,000

       2100 = 2 * 2 * 2 * 2 * 2 * …..14 times * ……… * 2 = 16,384 * ….. * 2

Since, 10,000 < 16,384 * ……. * 2

Thus, 2100 is greater than 1002.

(v) 210 = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 1,024

      102 = 10 * 10 = 100

Since, 1,024 > 100

Thus, 210 is greater than 102

Question 5:

Express each of the following as product of powers of their prime factors:

(i) 648          (ii) 405           (iii) 540           (iv) 3,600

Answer:

(i) 648 = 23 * 34

    Class_7_Exponents_&_Powers_Primefactorization_of_648                                         

(ii) 405 = 5 * 34           

   Class_7_Exponents_&_Powers_Primefactorization_of_405                                          

(iii) 540 = 22 * 33 * 5         

             Class_7_Exponents_&_Powers_Primefactorization_of_540                               

(iv) 3,600 = 24 * 32 * 52

  Class_7_Exponents_&_Powers_Primefactorization_of_3600                                         

Question 6:

Simplify:

(i) 2 * 103           (ii) 72 * 22             (iii) 23 * 5            (iv) 3 * 44                 (v) 0 * 102

(vi) 52 * 33              (vii) 24 * 32       (viii) 32 * 104

Answer:

(i) 2 * 103 = 2 * 10 * 10 * 10 = 2,000

(ii) 72 * 22 = 7 * 7 * 2 * 2 = 196

(iii) 23 * 5 = 2 * 2 * 2 * 5 = 40

(iv) 3 * 44 = 3 * 4 * 4 * 4 * 4 = 768

(v) 0 * 102 = 0 * 10 * 10 = 0

(vi) 52 * 33 = 5 * 5 * 3 * 3 * 3 = 675

(vii) 24 * 32 = 2 * 2 * 2 * 2 * 3 * 3 = 144

(viii) 32 * 104 = 3 * 3 * 10 * 10 * 10 * 10 = 90,000

Question 7:

Simplify:

(i) (-4)3          (ii) (-3) * (-2)3           (iii) (-3)2 * (-5)2           (iv) (-2)3 * (-10)3

Answer:

(i) (-4)3 = (-4) * (-4) * (-4) = -64         

(ii) (-3) * (-2)3 = (-3) * (-2) * (-2) * (-2) = 24         

(iii) (-3)2 * (-5)2 = (-3) * (-3) * (-5) * (-5) = 225         

(iv) (-2)3 * (-10)3 = (-2) * (-2) *(-2) *(-10) *(-10) *(-10) = 8000

Question 8:

Compare the following numbers:

(i) 2.7 * 1012; 1.5 * 108               (ii) 4 * 1014; 3 * 1017

Answer:

(i) 2.7 * 1012 and 1.5 * 108

On comparing the exponents of base 10,

2.7 * 1012 > 1.5 * 108

(ii) 4 * 1014  and 3 * 1017

On comparing the exponents of base 10,

4 * 1014 < 3 * 1017

                                                               Exercise 13.2

Question 1:

Using laws of exponents, simplify and write the answer in exponential form:

(i) 32 * 34 * 38               (ii) 615/610                (iii) a3 * a2             (iv) 7x * 72          (v) (52)3 /53              

(vi) 25 * 55                   (vii) a4 * b4                (viii) (34)3              (ix) (220/215) * 23        (x) 8t/82

Answer:

(i) 32 * 34 * 38 = 32+4+8 = 314                                                     [Since am * an = am+n ]          

(ii) 615/610 = 615-10 = 65                                                             [Since am / an = am-n ]              

(iii) a3 * a2 = a3+2 = a5                                                               [Since am * an = am+n ]           

(iv) 7x * 72 = 7x+2                                                                       [Since am * an = am+n ]

(v) (52)3 /53 = 56 /53 =  56-3 = 53                                               [Since (am)n = amn and am * an = am+n ]

(vi) 25 * 55 = (2 * 5)5 = 105                                                      [Since am * bm = (a * b)m ]

(vii) a4 * b4 = (a * b)4                                                               [Since am * bm = (a * b)m ]

(viii) (34)3 = 312                                                                          [Since (am)n = amn ]

(ix) (220/215) * 23 = 220-15 * 23 = 25 * 23 

                                                   = 25+3 = 28                                [Since am / an = am-n and am * an = am+n ]

(x) 8t/82 = 8t-2                                                                                                                       [Since am / an = am-n  ]

Question 2:

Simplify and express each of the following in exponential form:

(i) (23 * 34 * 4)/(3 * 32)          (ii) [(52)3 * 54]/57         (iii) 254 / 53            (iv) (3 * 72 * 118)/(21 * 113)

(v) 37/(34 * 33)      (vi) 20 + 30 + 40     (vii) 20 * 30 * 40   (viii) (30 + 20) * 50    (ix) (28 * a5)/(43 * a3)   

(x) (a5/a3) * a8         (xi) (45 * a8 b3)/ (45 * a5 b2)    (xii) (23 * 2)2

 

Answer:

(i) (23 * 34 * 4)/(3 * 32) = (23 * 34 * 22)/(3 * 25)      

                                          = (23+2 * 34)/(3 * 25)

                                          = (25 * 34)/(3 * 25)

                                          = 25-5 * 34-1

                                          = 20 * 33

                                          = 1 * 33

                                          = 33 

(ii) [(52)3 * 54]/57 = [56 * 54]/57

                              = 56+4/57

                              = 510/57

                              = 510-7

                              = 53  

(iii) 254 / 53 = (52)4 / 53          

                      = 58 / 53

                      = 58-3  

                     = 55

(iv) (3 * 72 * 118)/(21 * 11) = (3 * 72 * 118)/(3 * 7 * 113)

                                                = 31-1 * 72-1 * 118-3

                                                = 30 * 7 * 115 

                                                = 1 * 7 * 115

                                                = 7 * 115         

 (v) 37/(34 * 33) = 37/34+3

                        = 37/37

                        = 30   

                        = 1           

(vi) 20 + 30 + 40 = 1 + 1 + 1 = 3   

(vii) 20 * 30 * 40 = 1 * 1 * 1 = 1 

(viii) (30 + 20) * 50 = (1 + 1) * 1 = 2 * 1 = 2 

(ix) (28 * a5)/(43 * a3) = (28 * a5)/{(22)3 * a3)}  

                                      = (28 * a5)/(26 * a3)}

                                      = 28-6 * a5-3

                                      = 22 * a2  

                                      = 4a2

                                      = (2a)2

(x) (a5/a3) * a8 = a5-3 * a8       

                         = a2 * a8

                         = a2+8

                         = a10

(xi) (45 * a8 b3)/ (45 * a5 b2) = 45-5 * a8-5 b3-2   

                                                = 40 * a3 b

                                                = 1 * a3 b

                                                = a3 b

(xii) (23 * 2)2 = (23+1)2 = (24)2 = 24*2 = 28

 

 

Question 3:

Say true or false and justify your answer:

(i) 10 * 1011 = 10011       (ii) 23 > 52       (iii) 23 * 32 = 65        (iv) 30 = (1000)0

Answer:

(i) 10 * 1011 = 10011

LHS: 10 * 1011 = 1011+1 = 1012 =

RHS: 10011 = (102)11 = 102*11 = 1022

Since LHS ≠ RHS

So, it is false.

 

(ii) 23 > 52   

LHS: 23 = 8

RHS: 52 = 25

Since LHS is not greater than RHS

Therefore, it is false.

 

(iii) 23 * 32 = 65

LHS: 23 * 32 = 8 * 9 = 72

RHS: 65 = 7,776

Since LHS ≠ RHS

Therefore, it is false.

 

(iv) 30 = (1000)0

LHS: 30 = 1 and RHS: (1000)0 = 1

Since LHS = RHS

Therefore, it is true.

Question 4:

Express each of the following as a product of prime factors only in exponential form:

(i) 108 * 192        (ii) 270          (iii) 729 * 64         (iv) 768

Answer:

(i) 108 x 192 = (22 * 33)*(26 * 3)

                       = 22+6 * 33+1

                       = 28 * 34

 

 Class_7_Exponents_&_Powers_Primefactorization1    Class_7_Exponents_&_Powers_Primefactorization2

 

 

(ii) 270 = 2 * 35 * 5

 Class_7_Exponents_&_Powers_Primefactorization_of_270

 

 

(iii) 729 x 64 = 36 * 26

 Class_7_Exponents_&_Powers_Primefactorization_of_729    Class_7_Exponents_&_Powers_Primefactorization_of_64

 

(iv) 768 = 28 * 3

       Class_7_Exponents_&_Powers_Primefactorization_of_768                                          

Question 5:

Simplify:

(i) {(25)2 * 73}/(83 * 7)         (ii) (25 * 52 * t8)/(103 * t4)         (iii) (35 * 105 * 25)/(57 * 65)

Answer:

(i) {(25)2 * 73}/(83 * 7) = (25*2 * 73)/{(23)3 * 7)} 

                                       = (210 * 73)/(23*3 * 7)

                                       = (210 * 73)/(29 * 7)

                                       = 210-9 * 73-1

                                       = 2 * 72

                                       = 2 * 49

                                       = 98   

(ii) (25 * 52 * t8)/(103 * t4) =(52 * 52 * t8)/{(5 * 2)3 * t4}

                                               = (52+2 * t8)/(53 * 23 * t4)

                                               = (54 * t8)/(53 * 23 * t4)  

                                               = (54-3 * t8-4)/8

                                             = 5t4/8

(iii) (35 * 105 * 25)/(57 * 65) = {(35 * (2 * 5)5 * 52)}/{57 * (2 * 3)5}

                                                = (35 * 25 * 55+2)/(57 * 25 * 35)

                                                = (35 * 25 * 57)/(57 * 25 * 35)

                                                = 35-5 * 25-5 * 57-7

                                                = 30 * 20 * 50

                                                = 1 * 1 * 1

                                                = 1

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