## EXERCISE 2.1 PAGE NO: 2.8

1. Express each of the following as a rational number of the form p/q, where p and q are integers and q ≠ 0:
(i) 2-3
(ii) (-4)-2
(iii) 1/(3)-2
(iv) (1/2)-5
(v) (2/3)-2
Solution:
(i) 2-3 = 1/23 = 1/2×2×2 = 1/8 (we know that a-n = 1/an)
(ii) (-4)-2 = 1/-42 = 1/-4×-4 = 1/16 (we know that a-n = 1/an)
(iii) 1/(3)-2 = 32 = 3×3 = 9 (we know that 1/a-n = an)
(iv) (1/2)-5 = 2/ 15 = 2×2×2×2×2 = 32 (we know that a-n = 1/an)
(v) (2/3)-2 = 32 / 22 = 3×3 / 2×2 = 9/4 (we know that a-n = 1/an)
2. Find the values of each of the following:
(i) 3-1 + 4-1
(ii) (30 + 4-1) × 22
(iii) (3-1 + 4-1 + 5-1)0
(iv) ((1/3)-1 – (1/4)-1)-1
Solution:
(i) 3-1 + 4-1
1/3 + 1/4 (we know that a-n = 1/an)
LCM of 3 and 4 is 12
(1×4 + 1×3)/12
(4+3)/12
7/12
(ii) (30 + 4-1) × 22
(1 + 1/4) × 4 (we know that a-n = 1/an, a0 = 1)
LCM of 1 and 4 is 4
(1×4 + 1×1)/4 × 4
(4+1)/4 × 4
5/4 × 4
5
(iii) (3-1 + 4-1 + 5-1)0
(We know that a0 = 1)
(3-1 + 4-1 + 5-1)0 = 1
(iv) ((1/3)-1 – (1/4)-1)-1
(31 – 41)-1 (we know that 1/a-n = an, a-n = 1/an)
(3-4)-1
(-1)-1
1/-1 = -1
3. Find the values of each of the following:
(i) (1/2)-1 + (1/3)-1 + (1/4)-1
(ii) (1/2)-2 + (1/3)-2 + (1/4)-2
(iii) (2-1 × 4-1) ÷ 2-2
(iv) (5-1 × 2-1) ÷ 6-1
Solution:
(i) (1/2)-1 + (1/3)-1 + (1/4)-1
21 + 31 + 41 (we know that 1/a-n = an)
2+3+4 = 9
(ii) (1/2)-2 + (1/3)-2 + (1/4)-2
22 + 32 + 42 (we know that 1/a-n = an)
2×2 + 3×3 + 4×4
4+9+16 = 29
(iii) (2-1 × 4-1) ÷ 2-2
(1/21 × 1/41) / (1/22) (we know that a-n = 1/an)
(1/2 × 1/4) × 4/1 (we know that 1/a ÷ 1/b = 1/a × b/1)
1/2
(iv) (5-1 × 2-1) ÷ 6-1
(1/51 × 1/21) / (1/61) (we know that a-n = 1/an)
(1/5 × 1/2) × 6/1 (we know that 1/a ÷ 1/b = 1/a × b/1)
3/5
4. Simplify:
(i) (4-1 × 3-1)2
(ii) (5-1 ÷ 6-1)3
(iii) (2-1 + 3-1)-1
(iv) (3-1 × 4-1)-1 × 5-1
Solution:
(i) (4-1 × 3-1)2 (we know that a-n = 1/an)
(1/4 × 1/3)2
(1/12)2
(1×1 / 12×12)
1/144
(ii) (5-1 ÷ 6-1)3
((1/5) / (1/6))3 (we know that a-n = 1/an)
((1/5) × 6)3 (we know that 1/a ÷ 1/b = 1/a × b/1)
(6/5)3
6×6×6 / 5×5×5
216/125
(iii) (2-1 + 3-1)-1
(1/2 + 1/3)-1 (we know that a-n = 1/an)
LCM of 2 and 3 is 6
((1×3 + 1×2)/6)-1
(5/6)-1
6/5
(iv) (3-1 × 4-1)-1 × 5-1
(1/3 × 1/4)-1 × 1/5 (we know that a-n = 1/an)
(1/12)-1 × 1/5
12/5
5. Simplify:
(i) (32 + 22) × (1/2)3
(ii) (32 – 22) × (2/3)-3
(iii) ((1/3)-3 – (1/2)-3) ÷ (1/4)-3
(iv) (22 + 32 – 42) ÷ (3/2)2
Solution:
(i) (32 + 22) × (1/2)3
(9 + 4) × 1/8 = 13/8
(ii) (32 – 22) × (2/3)-3
(9-4) × (3/2)3
5 × (27/8)
135/8
(iii) ((1/3)-3 – (1/2)-3) ÷ (1/4)-3
(3– 23) ÷ 43 (we know that 1/a-n = an)
(27-8) ÷ 64
19 × 1/64 (we know that 1/a ÷ 1/b = 1/a × b/1)
19/64
(iv) (22 + 32 – 42) ÷ (3/2)2
(4 + 9 – 16) ÷ (9/4)
(-3) × 4/9 (we know that 1/a ÷ 1/b = 1/a × b/1)
-4/3
6. By what number should 5-1 be multiplied so that the product may be equal to (-7)-1?
Solution:
Let us consider a number x
So, 5-1 × x = (-7)-1
1/5 × x = 1/-7
x = (-1/7) / (1/5)
= (-1/7) × (5/1)
= -5/7
7. By what number should (1/2)-1 be multiplied so that the product may be equal to (-4/7)-1?
Solution:
Let us consider a number x
So, (1/2)-1 × x = (-4/7)-1
1/(1/2) × x = 1/(-4/7)
x = (-7/4) / (2/1)
= (-7/4) × (1/2)
= -7/8
8. By what number should (-15)-1 be divided so that the quotient may be equal to (-5)-1?
Solution:
Let us consider a number x
So, (-15)-1 ÷ x = (-5)-1
1/-15 × 1/x = 1/-5
1/x = (1×-15)/-5
1/x = 3
x = 1/3

## EXERCISE 2.2 PAGE NO: 2.18

1. Write each of the following in exponential form:
(i) (3/2)-1 × (3/2)-1 × (3/2)-1 × (3/2)-1
(ii) (2/5)-2 × (2/5)-2 × (2/5)-2
Solution:
(i) (3/2)-1 × (3/2)-1 × (3/2)-1 × (3/2)-1
(3/2)-4 (we know that a-n = 1/an, an = a×a…n times)
(ii) (2/5)-2 × (2/5)-2 × (2/5)-2
(2/5)-6 (we know that a-n = 1/an, an = a×a…n times)
2. Evaluate:
(i) 5-2
(ii) (-3)-2
(iii) (1/3)-4
(iv) (-1/2)-1
Solution:
(i) 5-2
1/52 = 1/25 (we know that a-n = 1/an)
(ii) (-3)-2
(1/-3)2 = 1/9 (we know that a-n = 1/an)
(iii) (1/3)-4
34 = 81 (we know that 1/a-n = an)
(iv) (-1/2)-1
-21 = -2 (we know that 1/a-n = an)
3. Express each of the following as a rational number in the form p/q:
(i) 6-1
(ii) (-7)-1
(iii) (1/4)-1
(iv) (-4)-1 × (-3/2)-1
(v) (3/5)-1 × (5/2)-1
Solution:
(i) 6-1
1/61 = 1/6 (we know that a-n = 1/an)
(ii) (-7)-1
1/-71 = -1/7 (we know that a-n = 1/an)
(iii) (1/4)-1
41 = 4 (we know that 1/a-n = an)
(iv) (-4)-1 × (-3/2)-1
1/-41 × (2/-3)1 (we know that a-n = 1/an, 1/a-n = an)
1/-2 × -1/3
1/6
(v) (3/5)-1 × (5/2)-1
(5/3)1 × (2/5)1
5/3 × 2/5
2/3
4. Simplify:
(i) (4-1 × 3-1)2
(ii) (5-1 ÷ 6-1)3
(iii) (2-1 + 3-1)-1
(iv) (3-1 × 4-1)-1 × 5-1
(v) (4-1 – 5-1) ÷ 3-1
Solution:
(i) (4-1 × 3-1)2
(1/4 × 1/3)2 (we know that a-n = 1/an)
(1/12)2
1/144
(ii) (5-1 ÷ 6-1)3
(1/5 ÷ 1/6)3 (we know that a-n = 1/an)
(1/5 × 6)3 (we know that 1/a ÷ 1/b = 1/a × b/1)
(6/5)3
216/125
(iii) (2-1 + 3-1)-1
(1/2 + 1/3)-1 (we know that a-n = 1/an)
LCM of 2 and 3 is 6
((3+2)/6)-1
(5/6)-1 (we know that 1/a-n = an)
6/5
(iv) (3-1 × 4-1)-1 × 5-1
(1/3 × 1/4)-1 × 1/5 (we know that a-n = 1/an)
(1/12)-1 × 1/5 (we know that 1/a-n = an)
12 × 1/5
12/5
(v) (4-1 – 5-1) ÷ 3-1
(1/4 – 1/5) ÷ 1/3 (we know that a-n = 1/an)
LCM of 4 and 5 is 20
(5-4)/20 × 3/1 (we know that 1/a ÷ 1/b = 1/a × b/1)
1/20 × 3
3/20
5. Express each of the following rational numbers with a negative exponent:
(i) (1/4)3
(ii) 35
(iii) (3/5)4
(iv) ((3/2)4)-3
(v) ((7/3)4)-3
Solution:
(i) (1/4)3
(4)-3 (we know that 1/an = a-n)
(ii) 35
(1/3)-5 (we know that 1/an = a-n)
(iii) (3/5)4
(5/3)-4 (we know that (a/b)-n = (b/a)n)
(iv) ((3/2)4)-3
(3/2)-12 (we know that (an)m = anm)
(v) ((7/3)4)-3
(7/3)-12 (we know that (an)m = anm)
6. Express each of the following rational numbers with a positive exponent:
(i) (3/4)-2
(ii) (5/4)-3
(iii) 43 × 4-9
(iv) ((4/3)-3)-4
(v) ((3/2)4)-2
Solution:
(i) (3/4)-2
(4/3)2 (we know that (a/b)-n = (b/a)n)
(ii) (5/4)-3
(4/3)3 (we know that (a/b)-n = (b/a)n)
(iii) 43 × 4-9
(4)3-9 (we know that an × am = an+m)
4-6
(1/4)6 (we know that 1/an = a-n)
(iv) ((4/3)-3)-4
(4/3)12 (we know that (an)m = anm)
(v) ((3/2)4)-2
(3/2)-8 (we know that (an)m = anm)
(2/3)8 (we know that 1/an = a-n)
7. Simplify:
(i) ((1/3)-3 – (1/2)-3) ÷ (1/4)-3
(ii) (32 – 22) × (2/3)-3
(iii) ((1/2)-1 × (-4)-1)-1
(iv) (((-1/4)2)-2)-1
(v) ((2/3)2)3 × (1/3)-4 × 3-1 × 6-1
Solution:
(i) ((1/3)-3 – (1/2)-3) ÷ (1/4)-3
(33 – 23) ÷ 43 (we know that 1/an = a-n)
(27-8) ÷ 64
19 ÷ 64
19 × 1/64 (we know that 1/a ÷ 1/b = 1/a × b/1)
19/64
(ii) (32 – 22) × (2/3)-3
(9 – 4) × (3/2)3 (we know that 1/an = a-n)
5 × (27/8)
135/8
(iii) ((1/2)-1 × (-4)-1)-1
(21 × (1/-4))-1 (we know that 1/an = a-n)
(1/-2)-1 (we know that 1/an = a-n)
-21
-2
(iv) (((-1/4)2)-2)-1
((-1/16)-2)-1 (we know that 1/an = a-n)
((-16)2)-1 (we know that 1/an = a-n)
(256)-1 (we know that 1/an = a-n)
1/256
(v) ((2/3)2)3 × (1/3)-4 × 3-1 × 6-1
(4/9)3 × 34 × 1/3 × 1/6 (we know that 1/an = a-n)
(64/729) × 81 × 1/3 × 1/6
(64/729) × 27 × 1/6
32/729 × 27 × 1/3
32/729 × 9
32/81
8. By what number should 5-1 be multiplied so that the product may be equal to (-7)-1?
Solution:
Let us consider a number x
So, 5-1 × x = (-7)-1
1/5 × x = 1/-7 (we know that 1/an = a-n)
x = (-1/7) / (1/5)
= (-1/7) × (5/1) (we know that 1/a ÷ 1/b = 1/a × b/1)
= -5/7
9. By what number should (1/2)-1 be multiplied so that the product may be equal to (-4/7)-1?
Solution:
Let us consider a number x
So, (1/2)-1 × x = (-4/7)-1
1/(1/2) × x = 1/(-4/7) (we know that 1/an = a-n)
x = (-7/4) / (2/1)
= (-7/4) × (1/2) (we know that 1/a ÷ 1/b = 1/a × b/1)
= -7/8
10. By what number should (-15)-1 be divided so that the quotient may be equal to (-5)-1?
Solution:
Let us consider a number x
So, (-15)-1 ÷ x = (-5)-1 (we know that 1/a ÷ 1/b = 1/a × b/1)
1/-15 × 1/x = 1/-5 (we know that 1/an = a-n)
1/x = (1×-15)/-5
1/x = 3
x = 1/3
11. By what number should (5/3)-2 be multiplied so that the product may be (7/3)-1?
Solution:
Let us consider a number x
So, (5/3)-2 × x = (7/3)-1
1/(5/3)2 × x = 1/(7/3) (we know that 1/an = a-n)
x = (3/7) / (3/5)2
= (3/7) / (9/25)
= (3/7) × (25/9) (we know that 1/a ÷ 1/b = 1/a × b/1)
= (1/7) × (25/3)
= 25/21
12. Find x, if
(i) (1/4)-4 × (1/4)-8 = (1/4)-4x
Solution:
(1/4)-4 × (1/4)-8 = (1/4)-4x
(1/4)-4-8 = (1/4)-4x (we know that an × am = an+m)
(1/4)-12 = (1/4)-4x
When the bases are same we can directly equate the coefficients
-12 = -4x
x = -12/-4
= 3
(ii) (-1/2)-19 ÷ (-1/2)8 = (-1/2)-2x+1
Solution:
(-1/2)-19 ÷ (-1/2)8 = (-1/2)-2x+1
(1/2)-19-8 = (1/2)-2x+1 (we know that an ÷ am = an-m)
(1/2)-27 = (1/2)-2x+1
When the bases are same we can directly equate the coefficients
-27 = -2x+1
-2x = -27-1
x = -28/-2
= 14
(iii) (3/2)-3 × (3/2)5 = (3/2)2x+1
Solution:
(3/2)-3 × (3/2)5 = (3/2)2x+1
(3/2)-3+5 = (3/2)2x+1 (we know that an × am = an+m)
(3/2)2 = (3/2)2x+1
When the bases are same we can directly equate the coefficients
2 = 2x+1
2x = 2-1
x = 1/2
(iv) (2/5)-3 × (2/5)15 = (2/5)2+3x
Solution:
(2/5)-3 × (2/5)15 = (2/5)2+3x
(2/5)-3+15 = (2/5)2+3x (we know that an × am = an+m)
(2/5)12 = (2/5)2+3x
When the bases are same we can directly equate the coefficients
12 = 2+3x
3x = 12-2
x = 10/3
(v) (5/4)-x ÷ (5/4)-4 = (5/4)5
Solution:
(5/4)-x ÷ (5/4)-4 = (5/4)5
(5/4)-x+4 = (5/4)5 (we know that an ÷ am = an-m)
When the bases are same we can directly equate the coefficients
-x+4 = 5
-x = 5-4
-x = 1
x = -1
(vi) (8/3)2x+1 × (8/3)5 = (8/3) x+2
Solution:
(8/3)2x+1 × (8/3)5 = (8/3)x+2
(8/3)2x+1+5 = (8/3) x+2 (we know that an × am = an+m)
(8/3)2x+6 = (8/3) x+2
When the bases are same we can directly equate the coefficients
2x+6 = x+2
2x-x = -6+2
x = -4
13. (i) If x= (3/2)2 × (2/3)-4, find the value of x-2.
Solution:
x= (3/2)2 × (2/3)-4
= (3/2)2 × (3/2)4 (we know that 1/an = a-n)
= (3/2)2+4 (we know that an × am = an+m)
= (3/2)6
x-2 = ((3/2)6)-2
= (3/2)-12
= (2/3)12
(ii) If x = (4/5)-2 ÷ (1/4)2, find the value of x-1.
Solution:
x = (4/5)-2 ÷ (1/4)2
= (5/4)2 ÷ (1/4)2 (we know that 1/an = a-n)
= (5/4)2 × (4/1)2 (we know that 1/a ÷ 1/b = 1/a × b/1)
= 25/16 × 16
= 25
x-1 = 1/25
14. Find the value of x for which 52x ÷ 5-3 = 55
Solution:
52x ÷ 5-3 = 55
52x+3 = 55 (we know that an ÷ am = an-m)
When the bases are same we can directly equate the coefficients
2x+3 = 5
2x = 5-3
2x = 2
x = 1

## EXERCISE 2.3 PAGE NO: 2.22

1. Express the following numbers in standard form:
(i) 6020000000000000
Solution:
To express 6020000000000000 in standard form, count the total digits leaving 1st digit from the left. So the total number of digits becomes the power of 10. Therefore the decimal comes after the 1st digit.
the total digits leaving 1st digit from the left is 15
∴ the standard form is 6.02 × 1015
(ii) 0.00000000000942
Solution:
To express 0.00000000000942 in standard form,
Any number after the decimal point the powers become negative. Total digits after decimal is 12
∴ the standard form is 9.42 × 10-12
(iii) 0.00000000085
Solution:
To express 0.00000000085 in standard form,
Any number after the decimal point the powers become negative. Total digits after decimal is 10
∴ the standard form is 8.5 × 10-10
(iv) 846 × 107
Solution:
To express 846 × 107 in standard form, count the total digits leaving 1st digit from the left. So the total number of digits becomes the power of 10. Therefore the decimal comes after the 1st digit.
the total digits leaving 1st digit from the left is 2
846 × 10= 8.46 × 10× 107 = 8.46 × 102+7 = 8.46 × 109
(v) 3759 × 10-4
Solution:
To express 3759 × 10-4 in standard form, count the total digits leaving 1st digit from the left. So the total number of digits becomes the power of 10. Therefore the decimal comes after the 1st digit.
the total digits leaving 1st digit from the left is 3
3759 × 10-4 = 3.759 × 10× 10-4 = 3.759 × 103+(-4) = 3.759 × 10-1
(vi) 0.00072984
Solution:
To express 0.00072984 in standard form,
Any number after the decimal point the powers become negative. Total digits after decimal is 4
∴ the standard form is 7.2984 × 10-4
(vii) 0.000437 × 104
Solution:
To express 0.000437 × 10in standard form,
Any number after the decimal point the powers become negative. Total digits after decimal is 4
∴ the standard form is 4.37 × 10-4 × 104 = 4.37
(viii) 4 ÷ 100000
Solution:
To express in standard form count the number of zeros of the divisor. This count becomes the negative power of 10.
∴ the standard form is 4 × 10-5
2. Write the following numbers in the usual form:
(i) 4.83 × 107
Solution:
When the powers are positive the usual form of number is written after the multiplication of the given number, then place the decimal point after counting from right.
4.83 × 10000000 = 4830000000
48300000.00
∴ the usual form is 48300000
(ii) 3.02 × 10-6
Solution:
When the powers are negative the decimal is placed to the left of the number.
3.02 × 10-6 here, the power is -6, so the decimal shifts 6 places to left.
∴ the usual form is 0.00000302
(iii) 4.5 × 104
Solution:
When the powers are positive the usual form of number is written after the multiplication of the given number, then place the decimal point after counting from right.
4.5 × 10000 = 450000
45000.0
∴ the usual form is 45000
(iv) 3 × 10-8
Solution:
When the powers are negative the decimal is placed to the left of the number.
3 × 10-6 here, the power is -8, so the decimal shifts 8 places to left.
∴ the usual form is 0.00000003
(v) 1.0001 × 109
Solution:
When the powers are positive the usual form of number is written after the multiplication of the given number, then place the decimal point after counting from right.
1.0001 × 1000000000 = 10001000000000
1000100000.0000
∴ the usual form is 1000100000
(vi) 5.8 × 102
Solution:
When the powers are positive the usual form of number is written after the multiplication of the given number, then place the decimal point after counting from right.
5.8 × 100 = 5800
580.0
∴ the usual form is 580
(vii) 3.61492 × 106
Solution:
When the powers are positive the usual form of number is written after the multiplication of the given number, then place the decimal point after counting from right.
3.61492 × 1000000 = 361492000000
3614920.00000
∴ the usual form is 3614920
(vii) 3.25 × 10-7
Solution:
When the powers are negative the decimal is placed to the left of the number.
3.25 × 10-7 here, the power is -7, so the decimal shifts 7 places to left.
∴ the usual form is 0.000000325
Courtesy : CBSE