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Page No 2.18:

Question 1:

Write each of the following in exponential form:
(i) 32-1×32-1×32-1×32-1
(ii) 25-2×25-2×25-2

Answer:

(i) 32-1×32-1×32-1×32-1=32-1+-1+-1+-1                                am×an=am+n  =32-4(ii) 25-2×25-2×25-2=25-2+-2+-2                                                                  am×an=am+n=25-6                                                                                    

Page No 2.18:

Question 2:

Evaluate:
(i) 5−2
(ii) (−3)−2
(iii) 13-4
(iv) -12-1

Answer:

(i) 5-2=152              ---> (an = 1/(an))
              =125

(ii) (-3)2=132              ---> (an = 1/(an))
                        =19

(iii) 13-4=11/34              ---> (an = 1/(an))
                          =11/81
                          = 81

(iv) -12-1= 1-1/2         ---> (a−1 = 1/(a))
                              =-2
                             

Page No 2.18:

Question 3:

Express each of the following as a rational number in the form pq:
(i) 6−1
(ii) (−7)−1
(iii) 14-1
(iv) (-4)-1×-32-1
(v) 35-1×52-1

Answer:

(i) 6-1=16                ---> (a−1 = 1/a)

(ii) (-7)-1=1-7             ---> (a−1 = 1/a)
                        =-17

(iii) 14-1=11/4             ---> (a−1 = 1/a)
                          =4


(iv)  (-4)-1×-32-1=1-4×1-3/2           ---> (a−1 = 1/a)
                                                  =1-4×2-3
                                                  =16

(v)left(frac{3}{5} right )^{-1}times left(frac{5}{2} right )^{-1}=frac{1}{3/5}times frac{1}{5/2}             ---> (a−1 = 1/a)
                                              =53×25
                                              =23

Page No 2.18:

Question 4:

Simplify:
(i) 4-1×3-12
(ii) 5-1÷6-13
(iii) 2-1+3-1-1
(iv) 3-1×4-1-1×5-1
(v) 4-1-5-1÷3-1

Answer:

(i)left(4^{-1}times3^{-1} right )^2=left(frac{1}{4}times frac{1}{3}right )^2              ---> (a−1 = 1/a)
                                  =1122
                                  =(1)2(12)2                         --->((a/b)n = (an)/(bn))
                                  =1144

(ii)left(5^{-1}div6^{-1} right )^3=left(frac{1}{5}divfrac{1}{6} right)^3         ---> (a−1 = 1/a)
                                    =653
                                    =216125                         --->((a/b)n = (an)/(bn))


 (iii) 2-1 + 3-1-1 = 12+ 13-1     ---> (a-1= 1/a)   = 56-1 = 65                ---> (a-1= 1/a)


(iv)left(3^{-1}times4^{-1} right )^{-1}times5^{-1}=left(frac{1}{3}timesfrac{1}{4} right )^{-1}timesfrac{1}{5}         ---> (a−1 = 1/a)
                                                     =112-1×15
                                                     =12×15                              ---> (a−1 = 1/a)
                                                     =125

(v)left(4^{-1}-5^{-1} right )div 3^{-1}=left(frac{1}{4}-frac{1}{5} right )div frac{1}{3}         ---> (a−1 = 1/a)
                                               =left(frac{5-4}{20} right )times 3
                                               =120×3

                                               =320

Page No 2.18:

Question 5:

Express each of the following rational numbers with a negative exponent:
(i) 143
(ii) 35
(iii) 354
(iv) 324-3
(v) 734-3

Answer:

i.   143=41-3     a-n = 1anii.     35=13-5    a-n = 1aniii.   354=53-4    a-n = 1aniv.  324-3=32-12     amn = amnv.  734-3=73-12     amn = amn



Page No 2.19:

Question 6:

Express each of the following rational numbers with a positive exponent:
(i) 34-2
(ii) 54-3
(iii) 43×4-9
(iv) 43-3-4
(v) 324-2

Answer:

(i) 34-2 = 432          ---> (a−1 = 1/a)
                                      
 (ii) 54-3    = 453                   ---> (a−1 = 1/a)  

(iii) 43×4-9 = 43-9 = 4-6= 146         ---> (am x an = am+n)

(iv) 43-3-4= 43-4×-3= 4312   ---> ((am)n = amn)

(v) 324-2= 324×-2=  32-8= 238 ---> ((am)n = amn)

                    

Page No 2.19:

Question 7:

Simplify:
(i) 13-3-12-3÷14-3
(ii) 32-22×23-3
(iii) 12-1×(-4)-1-1
(iv) -142-2-1
(v) 2323×13-4×3-1×6-1

Answer:

(i)left(left(frac{1}{3} right )^{-3}- left(frac{1}{2} right )^{-3}right )div left(frac{1}{4} right )^{-3}=left(frac{1}{(1/3)^3}- frac{1}{left (1/2 right )^3}right )divfrac{1}{(1/4)^{3}}            ---> (an = 1/(an))
                                                                          =left(frac{1}{(1/27)}- frac{1}{left (1/8 right )}right )divfrac{1}{(1/64)}
                                                                          =left(frac{27}{1}- frac{8}{1}right )div64
                                                                          =left(19right )timesfrac{1}{64}
                                                                          =frac{19}{64}

(ii)left(3^2-2^2right )times left(frac{2}{3} right )^{-3}=left(9-4 right )timesfrac{1}{(2/3)^3}           ---> (an = 1/(an))
                                                  =5timesfrac{1}{8/27}
                                                  =5timesfrac{27}{8}
                                                  =frac{135}{8}

(iii)left(left(frac{1}{2}right )^{-1}times left(-4 right )^{-1}right)^{-1}=left (left(frac{1}{1/2}right )times left(frac{1}{-4} right ) right )^{-1}           ---> (a−1 = 1/a)
                                                            =left (2times left (frac{1}{-4} right ) right )^{-1}
                                                            =left (frac{1}{-2} right )^{-1}
                                                            =frac{1}{1/(-2)}                                               ---> (a−1 = 1/a)
                                                            =-2

(iv)left(left(left(frac{-1}{4} right )^2 right )^{-2} right )^{-1}=left (left (frac{(-1)^2}{4^2} right )^{-2} right )^{-1}          --> ((a/b)n = an/(bn))
                                                    =left (left (frac{1}{16} right )^{-2} right )^{-1}                  ---> (an = 1/(an))
                                                    =left (left (frac{1}{(1/16)^2} right ) right )^{-1}
                                                    =left (frac{1}{(1/256)} right )^{-1}
                                                    =256^{-1}                                     ---> (a−1 = 1/a)
                                                    =frac{1}{256}

(v)left(left(frac{2}{3} right )^2 right )^3timesleft(frac{1}{3} right )^{-4}times3^{-1}times6^{-1} =left (frac{2^2}{3^2} right )^3timesfrac{1}{(1/3)^4}timesfrac{1}{3}timesfrac{1}{6}   ---> ((a/b)n = an/(bn)) and (an = 1/(an))
                                                                               =left (frac{4}{9} right )^3timesfrac{1}{(1/81)}timesfrac{1}{3}timesfrac{1}{6}
                                                                               =frac{4^3}{9^3}times81timesfrac{1}{18}                             ---> ((a/b)n = an/(bn))
                                                                              =frac{64}{729}times81timesfrac{1}{18}
                                                                               =frac{64}{9}timesfrac{1}{18}
                                                                               =64timesfrac{1}{162}
                                                                               =frac{64}{162}
                                                                               =frac{32}{81}

Page No 2.19:

Question 8:

By what number should 5−1 be multiplied so that the product may be equal to (−7)−1?

Answer:

Expressing in fraction form, we get:
5−1 = 1/5 (using the property a−1 = 1/a)
and
(−7)−1 = −1/7 (using the property a−1 = 1/a).
We have to find a number x such that
frac{1}{5}x=frac{-1}{7}
Multiplying both sides by 5, we get:
x=-frac{5}{7}
Hence, 5−1 should be multiplied by −5/7 to obtain (−7)−1.

Page No 2.19:

Question 9:

By what number should 12-1 be multiplied so that the product may be equal to -47-1?

Answer:

Expressing in fractional form, we get:
(1/2)−1 = 2,       ---> (a−1 = 1/a)
and
(−4/7)−1 = −7/4     ---> (a−1 = 1/a)
We have to find a number x such that
2x=-frac{7}{4}
Dividing both sides by 2, we get:
x=-frac{7}{8}
Hence, (1/2)−1should be multiplied by −7/8 to obtain (−4/7)−1.

Page No 2.19:

Question 10:

By what number should (−15)−1 be divided so that the quotient may be equal to (−5)−1?

Answer:

Expressing in fractional form, we get:
(−15)−1 = −1/15,      ---> (a−1 = 1/a)
and
(−5)−1 = −1/5           ---> (a−1 = 1/a)
We have to find a number x such that
-frac{1}{15}div x=-frac{1}{5}
Solving this equation, we get:
-frac{1}{15}times frac{1}{x}=-frac{1}{5}
          -frac{1}{15}=-frac{x}{5}
          frac{-5}{-15}=x
therefore x=frac{1}{3}
Hence, (−15)−1should be divided by 1/3 to obtain (−5)−1.

Page No 2.19:

Question 11:

By what number should 53-2 be multiplied so that the product may be 73-1?

Answer:

Expressing as a positive exponent, we have:
left (frac{5}{3} right )^{-2}=frac{1}{(5/3)^2}        ---> (a−1 = 1/a)
                =frac{1}{25/9}            ---> ((a/b)n = (an)/(bn))
                =frac{9}{25}
and
(7/3) 1 = 3/7.                ---> (a−1 = 1/a)
We have to find a number x such that
frac{9}{25}times x=frac{3}{7}
Multiplying both sides by 25/9, we get:
x=frac{3}{7}timesfrac{25}{9}=frac{1}{7}timesfrac{25}{3}=frac{25}{21}
Hence, (5/3)−2 should be multiplied by 25/21 to obtain (7/3)−1.

Page No 2.19:

Question 12:

Find x, if
(i) 14-4×14-8=14-4x
(ii) -12-19×-128=-12-2x+1
(iii) 32-3×325=322x+1
(iv) 25-3×2515=252+3x
(v) 54-x÷54-4=545
(vi) 832x+1×835=83x+2

Answer:

(i) We have:

left(frac{1}{4} right )^{-4}timesleft(frac{1}{4} right )^{-8}&=&left(frac{1}{4} right )^{-4x}
                    left(frac{1}{4} right )^{-12}right )&=&left(frac{1}{4} right )^{-4x}(am×an = am+n)
                            -12=-4x
                                  3=x
x = 3

(ii) We have:

left(frac{-1}{2} right )^{-19}timesleft(frac{-1}{2} right )^{8}=left(frac{-1}{2} right )^{-2x+1}
                      left(frac{-1}{2} right )^{-11}=left(frac{-1}{2} right )^{-2x+1}(am×an = am+n)
                                  -11=-2x+1
                                  -12=-2x
                                        6=x
x = 6

(iii) We have:

left(frac{3}{2} right )^{-3}timesleft(frac{3}{2} right )^{5}=left(frac{3}{2} right )^{2x+1}
                      left(frac{3}{2} right )^{2}=left(frac{3}{2} right )^{2x+1}
                                2=2x+1
                                1=2x
                                frac{1}{2}=x
x = 1/2

(iv) We have:

left(frac{2}{5} right )^{-3}timesleft(frac{2}{5} right )^{15}=left(frac{2}{5} right )^{2+3x}
                     left(frac{2}{5} right )^{12}=left(frac{2}{5} right )^{2+3x}
                              12=2+3x
                              10=3x
                             frac{10}{3}=x
x = 10/3

(v) We have:

left(frac{5}{4} right )^{-x}divleft(frac{5}{4} right )^{-4}=left(frac{5}{4} right )^{5}
                 left(frac{5}{4} right )^{-x+4}=left(frac{5}{4} right )^{5}
                     -x+4=5
                              -x=1
                                  x=-1
x = −1

(vi) We have:

left(frac{8}{3} right )^{2x+1}timesleft(frac{8}{3} right )^{5}=left(frac{8}{3} right )^{x+2}
                  left(frac{8}{3} right )^{2x+6}=left(frac{8}{3} right )^{x+2}
                        2x+6=x+2
                                  x=-4
x = −4

Page No 2.19:

Question 13:

(i) If x=322×23-4, find the value of x−2.
(ii) If x=45-2÷142, find the value of x−1.

Answer:

(i) First, we have to find x.

x = 322×23-4  = 322×324   = 326          --->(a−1 = 1/a)
   
   
Hence, x−2 is:

x-2 = 326-2 = 32-12  = 2312             --->(a−1 = 1/a)


(ii) First, we have to find x.
x=left(frac{4}{5} right )^{-2} div left(frac{1}{4}right)^{2}    ---> ((a/b)n = (an)/(bn))
    =left(frac{4^{-2}}{5^{-2}} right ) times4^2
    =frac{4^{0}}{5^{-2}}
    =frac{1}{5^{-2}}                            ---> (a0 = 1)
Hence, the value of x−1 is:
x^{-1}=left (frac{1}{5^{-2}} right )^{-1}
         =left (5^2 right )^{-1}          --->(a−1 = 1/a)
         =frac{1}{5^{2}}                  --->(a−1 = 1/a)

Page No 2.19:

Question 14:

Find the value of x for which 52x ÷ 5−3 = 55.

Answer:

We have:
5^{2x}div 5^{-3}=5^5
         5^{2x+3}=5^5          ---> a^mdiv a^n=a^{m-n}
      2x+3=5
              2x=2
                x=1

Hence, x is 1.



Page No 2.22:

Question 1:

Express the following numbers in standard form:
(i) 6020000000000000
(ii) 0.00000000000943
(iii) 0.00000000085
(iv) 846 × 107
(v) 3759 × 10−4
(vi) 0.00072984
(vii) 0.000437 × 104
(viii) 4 ÷ 100000

Answer:

To express a number in the standard form, move the decimal point such that there is only one digit to the left of the decimal point.
(i) 6020000000000000 = 6.02 x 1015      (The decimal point is moved 15 places to the left.)
(ii) 0.0000000000943 = 9.43 x 10−12     (The decimal point is moved 12 places to the right.)
(iii) 0.00000000085 = 8.5 x 10−10     (The decimal point is moved 10 places to the right.)
(iv) 846 x 107 = 8.46 x 102 x 107 = 8.46 x 109     (The decimal point is moved two places to the left.)
(v) 3759 x 10−4 = 3.759 x 103 x 10−4 = 3.759 x 10−1     (The decimal point is moved three places to the left.)
(vi) 0.00072984 = 7.984 x 10−4     (The decimal point is moved four places to the right.)
(vii) 0.000437 x 104 = 4.37 x 10−4 x 104 = 4.37 x 100 = 4.37     (The decimal point is moved four places to the right.)
(viii) 4/100000 = 4 x 100000−1 = 4 x 10−5     (Just count the number of zeros in 1,00,000 to determine the exponent of 10.)

Page No 2.22:

Question 2:

Write the following numbers in the usual form:
(i) 4.83 × 107
(ii) 3.02 × 10−6
(iii) 4.5 × 104
(iv) 3 × 10−8
(v) 1.0001 × 109
(vi) 5.8 × 102
(vii) 3.61492 × 106
(viii) 3.25 × 10−7

Answer:

(i) 4.83 x 107 = 4.83 x 1,00,00,000 = 4,83,00,000
(ii) 3.02 x 10−6 = 3.02/106 = 3.02/10,00,000 = 0.00000302
(iii) 4.5 x 104 = 4.5 x 10,000 = 45,000
(iv) 3 x 10−8 = 3/108 = 3/10,00,00,000 = 0.00000003
(v) 1.0001 x 109 = 1.0001 x 1,00,00,00,000 = 1,00,01,00,000
(vi) 5.8 x 102 = 5.8 x 100 = 580
(vii)  3.61492 x 106 = 3.61492 x 10,00,000 = 3614920
(viii) 3.25 x 10−7 = 3.25/107 = 3.25/1,00,00,000 = 0.000000325

Page No 2.22:

Question 1:

Square of -23 is
(a) -23
(b) 23
(c) -49
(d) 49

Answer:

(d) 4/9
To square a number is to raise it to the power of 2. Hence, the square of (−2/3) is
(-2)232 = 49      ---> ( (a/b)n =  (an)/(bn) )               

Page No 2.22:

Question 2:

Cube of -12 is
(a) 18
(b) 116
(c) -18
(d) -116

Answer:

(c) -1/8
The cube of a number is the number raised to the power of 3. Hence the cube of −1/2 is

(-1)323        ---> ( (a/b)n = (an)/(bn
  =-18



Page No 2.23:

Question 3:

Which of the following is not equal to -354?
(a) (-3)454
(b) 34(-5)4
(c) -3454
(d) -35×-35×-35×-35

Answer:

(c)  −(34/54)

-354 = (-3)454 = 34(-5)4 = -35×-35×-35×-35
It is not equal to -3454.

Page No 2.23:

Question 4:

Which  of the following is not reciprocal of 234?
(a) 324
(b) 23-4
(c) 32-4
(d) 3424

Answer:

(c) (3/2)−4
The reciprocal of left (frac{2}{3} right )^4is left (frac{3}{2} right )^4.
Therefore, option (a) is the correct answer.
Option (b) is just re-expressing the number with a negative exponent.
Option (d) is obtained by working out the exponent.
Hence,option (c) is not the reciprocal of  left (frac{2}{3} right )^4.

Page No 2.23:

Question 5:

Which of the following numbers is not equal to -827?
(a) 23-3
(b) -233
(c) -233
(d) -23×-23×-23

Answer:

(a) (2/3)-3

We can write frac{-8}{27} as frac{-2times(-2)times(-2)}{3times3times3}. It can be written in the forms given below.


frac{-2times(-2)times(-2)}{3times3times3}=-frac{2times2times2}{3times3times3}            ---> work out the minuses
                               =-frac{2}{3}timesfrac{2}{3}timesfrac{2}{3}
                               =-left (frac{2}{3} right )^3                    
Hence, option (b) is equal to frac{-8}{27}.

We can also write:
frac{-2times(-2)times(-2)}{3times3times3}=left (-frac{2}{3} right )timesleft (-frac{2}{3} right )timesleft (-frac{2}{3} right )
                               =left (-frac{2}{3} right )^3
Hence, option (c) is also equal to frac{-8}{27}.

We can also write:
frac{-2times(-2)times(-2)}{3times3times3}=left (-frac{2}{3} right )timesleft (-frac{2}{3} right )timesleft (-frac{2}{3} right )
Hence, option (d) is also equal to -827.

This leaves out option (a) as the one not equal to -827.

Page No 2.23:

Question 6:

23-5 is equal to
(a) -235
(b) 325
(c) 2x-53
(d) 23×5

Answer:

(b)325

Rearrange (2/3)−5 to get a positive exponent.
23-5=1235         a-n=1an=12535                       abn=anbn=3525=325

Page No 2.23:

Question 7:

-125×-123 is equal to
(a) -128
(b) -128
(c) 148
(d) -1215

Answer:

(a) (−1/2)8

We have:

left (frac{-1}{2} right )^5timesleft (frac{-1}{2} right )^3=left (frac{-1}{2} right )^{5+3}
                              =left (frac{-1}{2} right )^8

Page No 2.23:

Question 8:

-153÷-158 is equal to
(a) -155
(b) -1511
(c) (-5)5
(d) 155

Answer:

(c)  (−5)5

We have:

left(frac{-1}{5} right )^3 div left(frac{-1}{5} right )^8 =left(frac{-1}{5} right )^{3-8}
                                        =left(frac{-1}{5} right )^{-5}
                                        =frac{1}{(-1/5)^5}
                                        =frac{1}{((-1)^5/5^5)}
                                        =frac{5^5}{(-1)^5}
                                        =left (frac{5}{-1} right )^5
                                        =left(-5 right )^5

Page No 2.23:

Question 9:

-257÷-255 is equal to
(a) 425
(b) -425
(c) -2512
(d) 254

Answer:

(a) 4/25

We have:

left(frac{-2}{5} right )^7 div left(frac{-2}{5} right )^5 =left(frac{-2}{5} right )^{7-5}
                                        =left (frac{-2}{5} right )^{2}
                                        =frac{(-2)^2}{5^2}
                                        =frac{4}{25}

Page No 2.23:

Question 10:

1324 is equal to
(a) 136
(b) 138
(c) 1324
(d) 1316

Answer:

(b) (1/3)8

We have:

left(left(frac{1}{3} right )^2 right )^4=left(frac{1}{3} right )^{2times4}          ---> ( (am)n = amxn)
                       =left(frac{1}{3} right )^{8}



Page No 2.24:

Question 11:

150 is equal to
(a) 0
(b) 15
(c) 1
(d) 5

Answer:

(c) 1

We have:

left (frac{1}{5} right )^0=1         ---> (a0 = 1, for every non-zero rational number a.)

Page No 2.24:

Question 12:

-32-1 is equal to
(a) 23
(b) -23
(c) 32
(d) none of these

Answer:

(b)-frac{2}{3}
We have:

left (frac{-3}{2} right )^{-1}=frac{1}{(-3)/2}          --> (a−1 = 1/a)
                    =frac{2}{-3}
                   

Page No 2.24:

Question 13:

23-5×57-5 is equal to
(a) 23×57-10
(b) 23×57-5
(c) 23×5725
(d) 23×57-25

Answer:

(b)left(frac{2}{3}times frac{5}{7} right ) ^{-5}

We have:

left(frac{2}{3} right ) ^{-5}times left(frac{5}{7} right ) ^{-5} =left(frac{2}{3}times frac{5}{7} right ) ^{-5}          ---> ((a x b)n = an x bn)

Page No 2.24:

Question 14:

345÷535 is equal to

Answer:

(a)  34÷535

We have:

left(frac{3}{4}right )^5 div left(frac{5}{3}right )^5 =left(frac{3}{4}divfrac{5}{3} right )^5            ---> an ÷ bn=a÷bn

Page No 2.24:

Question 15:

For any two non-zero rational numbers a and b, a4 ÷ b4 is equal to
(a) (a ÷ b)1
(b) (a ÷ b)0
(c) (a ÷ b)4
(d) (a ÷ b)8

Answer:

(c)left(adiv b right )^4
This is one of the basic exponential formulae, i.e. left(adiv b right )^n = a^n div b^n.

Page No 2.24:

Question 16:

For any two rational numbers a and b, a5 × b5 is equal to
(a) (a × b)0
(b) (a × b)10
(c) (a × b)5
(d) (a × b)25

Answer:

(c) (a x b)5
an x bn = (a x b)n
Hence,
a5 x b5 = (a x b)5

Page No 2.24:

Question 17:

For a non-zero rational number a, a7 ÷ a12 is equal to
(a) a5
(b) a−19
(c) a−5
(d) a19

Answer:

(c) a−5
a^m div a^n = a^{m-n}
Hence,
a^7 div a^{12}=a^{7-12}=a^{-5}

Page No 2.24:

Question 18:

For a non zero rational number a, (a3)−2 is equal to
(a) a9
(b) a−6
(c) a−9
(d) a1

Answer:

(b) a−6

We have:

left (a^3 right )^{-2}=a^{3times(-2)}          ---> ((am)n = am x n)
               =a^{-6}



Page No 2.8:

Question 1:

Express each of the following as a rational number of the form pq, where p and q are integers and q ≠ 0.
(i) 2−3
(ii) (−4)−2
(iii) 13-2
(iv) 12-5
(v) 23-2

Answer:

We know that a-n = 1an. Therefore,

(i)
 2^{-3} = frac{1}{2^{3}}=frac{1}{8}

(ii)


(iii)
frac{1}{3^{-2}} = 3^2=9

(iv)
left (frac{1}{2} right )^{-5} = 2^5=32

(v)
left (frac{2}{3} right )^{-2} = left (frac{3}{2} right )^{2}=frac{9}{4}

Page No 2.8:

Question 2:

Fiind the value of each of the following:
(i) 3−1 + 4−1
(ii) (30 + 4−1) × 22
(iii) (3−1 + 4−1 + 5−1)0
(iv) 13-1-14-1-1

Answer:

(i) We know from the property of powers that for every natural number a, a−1 = 1/a. Then:
3^{-1}+4^{-1}=frac{1}{3}+frac{1}{4}          ---> (a−1 = 1/a)
                      =frac{4+3}{12}
                      =frac{7}{12}

(ii) We know from the property of powers that for every natural number a, a−1 = 1/a.
Moreover, a0 is 1 for every natural number a not equal to 0. Then:

30+4-1×22=1+14×4    as, a-1=1a; a0=1=54×4=5
                 

(iii) We know from the property of powers that for every natural number a, a−1 = 1/a.
Moreover, a0 is 1 for every natural number a not equal to 0. Then:
(3-1+4-1+5-1)=1          ---> (Ignore the expression inside the bracket and use a0 = 1 immediately.)

(iv) We know from the property of powers that for every natural number a, a−1 = 1/a. Then:
left (left (frac{1}{3} right )^{-1}-left (frac{1}{4} right )^{-1} right )^{-1}=left ( 3-4 right )^{-1}          ---> (a−1 = 1/a)
                                                  =left ( -1 right )^{-1}
                                                  =-1                       ---> (a−1 = 1/a)

Page No 2.8:

Question 3:

Find the value of each of the following:
(i) 12-1+13-1+14-1
(ii) 12-2+13-2+14-2
(iii) (2−1 × 4−1) ÷ 2−2
(iv) (5−1 × 2−1) ÷ 6−1

Answer:

(i)
12-1+13-1+14-1=11/2+11/3+11/4          --> (a−1 = 1/a)
                                                           =2+3+4
                                                           =12

(ii)
12-2+13-2+14-2=11/22+11/32+11/42        --> (an = 1/(an))
                                                         = 11/4+11/9+11/16              --> ((a/b)n = (an/bn))
                                                         = 4+9+16
                                                          =29

(iii)
(2-1×4-1)÷2-2=12×14÷122   --> (an = 1/(an))
           
                                       =18×4
                                       = 2

(iv)
(5-1×2-1)÷6-1=15×12÷16             --> (an = 1/(an))
                                       =110×6

                                       =35

Page No 2.8:

Question 4:

Simplify:
(i) 4-1×3-12
(ii) 5-1÷6-13
(iii) 2-1+3-1-1
(iv) 3-1×4-1-1×5-1

Answer:

(i)
left (4^{-1}times 3^{-1} right )^2=left (frac{1}{4}times frac{1}{3} right )^2          ---> (a−1 = 1/a)
                            =left (frac{1}{12} right )^2
                            =frac{1^2}{12^2}                          ---> ((a/b)n = (an)/(bn) )
                            =124

(ii)
left (5^{-1}div 6^{-1} right )^3=left (frac{1}{5}div frac{1}{6} right )^3          ---> (a−1 = 1/a)
                           =left (frac{1}{5}times6 right )^3
                            =  653
                            =  6353                         ---> ((a/b)n = (an)/(bn) )

                            = 216125

(iii)
(2-1+3-1)-1 =12+13-1          ---> (a−1 = 1/a)
                             =  56-1
                               =15/6                          ---> (a−1 = 1/a)
                               =65

(iv)
left (3^{-1}times 4^{-1} right )^{-1}times 5^{-1}=left (frac{1}{3}times frac{1}{4} right )^{-1}times frac{1}{5}          ---> (a−1 = 1/a)
                                            =left (frac{1}{12}right )^{-1}times frac{1}{5} 
                                            =125                                      ---> (a−1 = 1/a)

Page No 2.8:

Question 5:

Simplify:
(i) 32+22×123
(ii) 32-22×23-3
(iii) 13-3-12-3÷14-3
(iv) 22+32-42÷322

Answer:

(i)
(32+22)×123=(9+4)×18=138

(ii)
left ( 3^2-2^2 right )times left(frac{2}{3} right )^{-3}=left(9-4 right )times frac{1}{left (2/3 right )^{3}}              ---> (a−1=1/(an))
                                          =5times frac{1}{8/27}                              ---> ((a/b)n = (an)/(bn))
                                          =5×278
                                          =1358

(iii)
left( left(frac{1}{3} right )^{-3}-left(frac{1}{2} right )^{-3}right )div left( frac{1}{4}right )^{-3}=left(3^3-2^3 right )div 4^3            --->(a-n = 1/(an))
                                                                  = (27-8)÷64
                                                                   =19×164
                                                                   =1964


(iv)
(22+32-42)÷322=(4+9-16)×94                    ---> ((a/b)n = (an)/(bn))

                                                  =-3×94
                                                  =-274

Page No 2.8:

Question 6:

By what number should 5−1 be multiplied so that the product may be equal to (−7)−1?

Answer:

Using the property a−1 = 1/a for every natural number a, we have 5−1 = 1/5 and (−7)−1 = −1/7. We have to find a number x such that
15×x=-17
Multiplying both sides by 5, we get:
x=-57
Hence, the required number is −5/7.

Page No 2.8:

Question 7:

By what number should 12-1 be multiplied so that the product may be equal to -47-1?

Answer:

Using the property a−1 = 1/a for every natural number a, we have (1/2)−1 = 2 and (−4/7)−1 = −7/4. We have to find a number x such that
2x=-74
Dividing both sides by 2, we get:
x=-78
Hence, the required number is −7/8.

Page No 2.8:

Question 8:

By what number should (−15)−1 be divided so that the quotient may be equal to (−5)−1?

Answer:

Using the property a−1 = 1/a for every natural number a, we have (−15)−1 = −1/15 and (−5)−1 = −1/5. We have to find a number x such that
-115x1= -15or -115×1x= -15or x = 13
Hence, (−15)−1 should be divided by 13 to obtain (−5)−1.



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