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Mass of Uranus = 86,800,000,000,000,000,000,000,000 kg.
Distance between Sun and Saturn = 1,433,500,000,000 m
Distance between Saturn and Uranus = 1,439,000,000,000 m.
How would we read these numbers ?
Which is heavier, Earth or Uranus?
Which distance is less ? Between Sun & Saturn or between Saturn & Uranus?
These numbers are very large and hence difficult to read, understand and compare.
To make these numbers easy to read, understand and compare, we use exponents.
Exponents
We can express a larger number as the products of its factors as100000 = 100×1000
or 100000 = 100×50×20
or 100000 = 10×10×10×5×2×10
or 100000 = 10×10×10×10×10
In last expression, we can see that same number 10 is multiplied 5 times, we use notation 105 to represent this product 10×10×10×10×10.
100000 = 10×10×10×10×10 = 105
It is an exponential form of 100000 and is read as 10 raised to the power of 5.
Here ‘10’ is called the base and ‘5’ the exponent.
∴ An exponent refers to the number of times a number is multiplied by itself.
"Exponential form is a way of expressing a standard number using a base and a raised number called an exponent."
➤ Numbers in expanded form can be written in exponential form
25769 = 2 × 10000 + 5 × 1000 + 7 × 100 + 6 × 10 + 9
= 2 × 104 + 5 × 103 + 7 × 102 + 6 × 10 + 9
➤ Exponential numbers having base other than 10,
16 = 2 × 2 × 2 × 2 = 24 here, 2 is the base & 4 is the exponent.
16 = 4 × 4 = 42 here, 4 is the base & 2 is the exponent.
64 = 2 × 2 × 2 × 2 × 2 × 2 = 26 here, 2 is the base & 6 is the exponent.
64 = 4 ×4 × 4 = 43 here, 4 is the base & 3 is the exponent.64 = 2 × 2 × 2 × 2 × 2 × 2 = 26 here, 2 is the base & 6 is the exponent.
81 = 3 × 3 × 3 × 3 = 34 here, 3 is the base & 4 is the exponent.
Some exponents have special names-
↬ Square — When the exponent is 2 i.e. when the base is raised to 2, it is said to be squired.32, which is 3 raised to the power 2, also read as ‘3 squared’
32 = 3×3 =9
102, which is 10 raised to the power 2, also read as ‘10 squared’
102 = 10×10 =100
↬ Cube — When the exponent is 3 i.e. when the base is raised to 3, it is said to be cubed.
23, which is 2 raised to the power 3, also read as ‘2 cubed’
23 = 2×2×2 = 8
53, which is 5 raised to the power 3, also read as ‘5 cubed’
53 = 5×5×5 = 125
For any integer a
a = a1
a × a = a2 (read as ‘a squared’ or ‘a raised to the power 2’)
a × a × a = a3 (read as ‘a cubed’ or ‘a raised to the power 3’)
a × a × a × a = a4 (read as a raised to the power 4 or the 4th power of a)
..............................
a × a × a × a × a × a × a = a 7 (read as a raised to the power 7 or the 7th power of a)
and so on.
Exponential form of negative integer
↪ (−2)2 = (−2)×(−2) = 4,Value of a negative integer raised to even positive power is positive.
↪ (−2)3 = (−2)×(−2)×(−2) = −8,
Value of a negative integer raised to odd positive power is negative.
(-a)m = a, for any positive even number m
(-a)n = -a, for any positive odd number n
Exponential form as power of different factors
144 = 9×16 =3×3×4×4 = 32×42Order of factor doesn't matter
(Multiplication is commutative )
32×42 = 42×32
a × a × a × b × b can be expressed as a3b2(read as a cubed b squared)
a 3 b 2 = b 2 a 3 (commutative Property)
a 2 b 4 = b 4 a 2
Exponential form as power of prime factors.
(Each base must be a prime number).144 = 9×16 =3×3×4×4 = 3×3×2×2×2×2 = 32×24
Prime factorise the number first, then express it in exponential form
432 = 2 × 2 × 2 × 2 × 3 × 3 × 3
or 432 = 24× 33
Try These
(1) Express: (i) 729 as a power of 3 (ii) 128 as a power of 2(iii) 343 as a power of 7
(2) Express as product of powers of their prime factors:
(i) 648 (ii) 405 (iii) 540
(3) Compare the following numbers:
(i) 2.7 × 1012 ; 1.5 × 108 (ii) 4 × 1014 ; 3 × 1017
Answer
(1) (i) 729 =3×3×3×3×3×3 = 36(ii) 128 = 2×2×2×2×2×2×2 = 27
(iii) 343 = 7×7×7 = 73
(2) (i) 648 = 2×2×2×3×3×3×3 = 23×34
(ii) 405 = 3×3×3×3×5 = 34×5
(3) (i) 2.7 × 1012 > 1.5 × 108
(ii) 4 × 1014 < 3 × 1017
Laws of Exponents
Multiplying Powers with the same base
→ 22×25 =2×2×2×2×2×2×2 = 27 = 2 2+5→ (−3) 2×(−3) 3 = (−3)×(−3)×(−3)×(−3)×(−3)
= (−3)5 = (−3)2+3
→ a2×a3 = a× a× a× a× a =a 5 = a 2+3
For any non-zero integers a, where m and n are whole numbers.
am× an = a m+n
Dividing Powers with the same base
→ 25 ÷ 22 = 2×2 ×2×2×2/2×2= 23 = 2 5-2
→ (−3) 3 ÷(−3) 2 = (−3)×(−3)×(−3)/(−3)×(−3)
= (−3) = (−3)3−2
→ a4 ÷ a2 = a× a× a× a/ a× a
= a 2 = a 4-2
For any non-zero integers a, where m and n are whole numbers and m > n.
am ÷ an = a m-n
Power of a Power
→ (53)2 =53×53 = 53+3 = 56 = 53×2→ (32) 5= 32×32×32×32×32
= 32+2+2+2+2 = 310 = 32×5
→ (a2)3 = a2× a2× a2 =a 2+2+2
= a6= a 2×3
For any non-zero integers a, where m and n are whole numbers.
(am)n = a m×n
Multiplying Powers with the same exponent
→ 23×53 =2×2×2×5×5×5 = (2×5)×(2×5)×(2×5)= 10×10×10 = 103 = (2×5)3
→ (−3)2×(2)2 = (−3)×(−3)×2×2 = (−3×2)×(−3×2)
= (−6)×(−6) = (−6)2 =(−3×2)2
→ a2×b2 = a× a× b× b = (a× b)×(a× b) = (ab)2 = (a×b)2
For any non-zero integers a and b, where m is any whole number.
am×bm = (a×b) m
Dividing Powers with the same exponent
→ 22 ÷ 32 = 2×2 /3×3 = 22/32 = (2/3)2→ a4 ÷ b4 = a× a× a× a/b×b×b×b = a4/b4 = (a/b)4
For any non-zero integers a and b , where m is any whole number.
am ÷ bm = (a/b) m
Exponent Zero
→ 22 ÷ 22 = 2×2 /2×2 = 4/4 = 1,also, 22 ÷ 22 = 22-2 = 20 ( using laws of exponents )
⇒ 20 = 1
→ 52 ÷ 52 = 5×5 /5×5 = 25/25 = 1,
also, 52 ÷ 52 = 52-2 = 50 ( using laws of exponents )
⇒ 50 = 1
For any non-zero integer a, a0 = 1
| ➧ a0 = 1
➧ am × an = a m+n ➧ am ÷ an = a m-n ➧ (am)n = a m×n ➧ am × bm = (a×b) m ➧ am ÷ bm = (a/b) |
exponent 0 same base same base power of power same exponent same exponent |
Try These
(1) Simplify and write in exponential form:(i) 25 × 23 (ii) p3 × p2 (iii) 911 ÷ 97
(iv) 2015 ÷ 2013 (v) (7015)2 (vi) 43 × 23
(vii) 25 × b5 (viii) 25 ÷ b5 (ix) (-2)3 ÷ b3
(2) Simplify: (25)2× 73 / 83 × 7
Answer
| (1) (i) 25+3 = 28 (ii) p3+2 = p5 (iii) 911-7 = 94 (iv) 2015-13 = 202 (v) 7015×2 = 7030 (vi) (4 × 2)3 = 83 (vii) (2× b)5= (2b)5 (ii) (2 ÷ b)5= (2/b)5 (iii) {(-2) ÷ b}3= (-2/b)3 | (2) (25)2× 73 / 83 × 7 = 25×2× 73 / (23)3 × 7 = 210× 73 / 29 × 7 = 210/ 29 × 73/ 71 = 210-9 × 73-1 = 2× 72 = 2×49 = 98 |
Expansion of Decimal numbers in the Exponential form
25769 = 2 × 10000 + 5 × 1000 + 7 × 100 + 6 × 10 + 9 × 1= 2 × 104 + 5 × 103 + 7 × 102 + 6 × 101 + 9 × 100
Standard Index form of a number
→ |a| should be greater or equal to 1 and less than 10, i.e., −10 < a ≤ −1 and 1 ≤ a < 10.
e.g., 79 = 7.9× 101
790 = 7.9×102
7900 = 7.9× 103 and so on.
→ A very large number or very small number can be easily expressed with it.
e.g. 423,000,000,000,000 = 4.23×1014
A very small number uses negative powers.
e.g. 0.000000000692 = 6.92×10−10
Comparing numbers in Standard Index form
Now, we can answer the questions asked at beginning by expressing those large numbers into standard form,Mass of earth = 5,970,000,000,000,000,000,000,000 kg
= 5.97×1024 kg
= five point nine seven times ten raised to the power of twenty-four.
Mass of Uranus = 86,800,000,000,000,000,000,000,000 kg.
= 8.68 ×1025 kg
= eight point six hundred eight times ten raised to the power of twenty-five.
Simply by comparing the powers of 10 in the above two, we can tell that the mass of Uranus is greater than that of the Earth.
8.680 ×1025 kg > 5.97×1024 kg
∴ Uranus is heavier than earth.
Distance between Sun and Saturn = 1,433,500,000,000 m =1.4335×1012 m
Distance between Saturn and Uranus = 1,439,000,000,000 m.= 1.439×1012 m
Powers of 10 in both numbers are same and 1.4335 < 1.439
1.4335×1012 m < 1.439×1012 m
∴ Distance between Sun & Saturn is less the distance between Saturn and Uranus.
Try These
(1) Expand by expressing powers of 10 in the exponential form:(i) 172 (ii) 56,439
Answer
(1)(i) 172 = 1×100 + 7×10 + 2×1 = 1×102 + 7×101 + 2×100(ii) 56,439 = 5×10,000 + 6×1000 + 4×100 + 3×10 + 9×1
= 5×104 + 6×103 +4×102 + 3×101 + 9×100
| Properties of Rational Numbers | |