Do you know that the distance between the Sun and Earth is 1,49,60,00,00 km? That's a huge number!
These very large numbers are difficult to read, understand, and compare.
Let's look at the distance in another form:
Distance between the Sun and Earth: 1.496×10^{8} kilometers.
But what does 10^{8 }mean? Let's understand this along with the chapter.
Exponents are a way to express very large or very small numbers in a more compact and manageable form, making it easier to read, understand, and compare them.
To write large numbers in short form, we use exponents.
Here 8 is the base, 3 is the exponent, and 8^{3} is the exponential form of 512.
This can be read as “8 raised to the power of 3”.
Similarly, 10, 000 = 10 × 10 × 10 × 10 = 10^{4 }(10 raised to the power of 4).
10^{4} is called the exponential form of 10,000.
Example: Express 256 as a power 2.
Ans: We have 256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2. So we can say that 256 = 2^{8.}
Example: Which one is greater 2^{3} or 3^{2}?
Ans: We have, 2^{3} = 2 × 2 × 2 = 8 and 3 ^{2} = 3 × 3 = 9.
Since 9 > 8, so, 3^{2} is greater than 2^{3.}
Example: Expand a^{3}b^{2}, a^{2}b^{3}, b^{2}a^{3}, b^{3}a^{2} . Are they all same?
Ans: a^{3}b^{2} = a^{3} × b^{2} = (a × a × a) × (b × b) = a × a × a × b × b
a^{2}b^{3} = a^{2 }x b^{3} = (a × a) × (b × b × b)
b^{2}a^{3 }= b^{2} × a^{3} = (b × b) × (a × a × a)
b^{3}a^{2} = b^{3} x a^{2} = (b × b × b) × (a × a)
In the terms a^{3}b^{2} and a^{2}b^{3} the powers of a and b are different. Thus a^{3}b^{2} and a^{2}b^{3} are different. On the other hand, a^{3}b^{2} and b^{2}a^{3} are the same, since the powers of a and b in these two terms are the same. The order of factors does not matter. Thus, a^{3}b^{2} = a^{3}× b^{2} = b^{2} × a^{3} = b^{2}a^{3}
Similarly, a^{2}b^{3} and b^{3}a^{2} are the same.
Example: Express the following numbers as a product of powers of prime factors:
(i) 432
Ans: 432 = 2 × 216 = 2 × 2 × 108 = 2 × 2 × (2 × 54) = 2 × 2 × 2 × 2 × 27
= 2 × 2 × 2 × 2 × 3 × 9
= 2 × 2 × 2 × 2 × 3 × 3 × 3 or 432
= 2^{4} × 3^{3}
(ii) 16000
Ans:16,000 = 16 × 1000 = (2 × 2 × 2 × 2) ×1000 = 2^{4} ×10^{3}
(as 16 = 2 × 2 × 2 × 2)
= (2 × 2 × 2 × 2) x (2 × 2 × 2 × 5 × 5 × 5)
= 2^{4} × 2^{3} × 5^{3}
(Since 1000 = 2 × 2 × 2 × 5 × 5 × 5)
= (2 × 2 × 2 × 2 × 2 × 2 × 2 ) × (5 × 5 × 5) or, 16,000
= 2^{7} × 5^{3}
^{}
A number expressed in decimal notation is written as a single number made up of integer digits in the units, tens, hundreds, thousands, et cetera positions to the left of the decimal point and possibly also in the tenths, hundredths, and so on to the right of the decimal point.
Let's understand this by writing number 4872 in expanded form using powers of 10:
4872 =4000+800+70+2 = 4×1000 + 8×100 + 7×10 + 2×1
4872 = ( 1×10^{3 })+( 8×10^{2 })+( 7×10^{1 })+( 2×10^{0 })
= 4872
Any number can be expressed as a decimal number between 1.0 and 10.0
(including 1.0) multiplied by a power of 10. Such a form of a number is called its standard form or scientific notation.
For example: Speed of light in vacuum = 300000000 m/s = 3.0 × 108 m/s.
The distance between the Sun and Earth is 149, 600,000,000 m = 1.496 × 10^{11}m.
Example: Express the following numbers in the standard form:
(i) 5985.3
Ans: 5985.3 = 5.9853 × 1000 = 5.9853 × 103
(ii) 70,040,000,000
Ans: 70,040,000,000 = 7.004 × 10,000,000,000 = 7.004 × 1010
Example: Solve the following
(i) 2^{3} x 2^{2}
Ans: Numbers raised to the power of three are called cube numbers From the law of exponent
we know, p^{m} x p^{n} =p^{(m+n)}
Therefore, 2^{3} x 2^{2} = 2^{(3+2)} = 2^{5} = 2 x 2 x 2 x 2 x 2 = 32
(ii) (5^{2})^{2}
Ans: We know, by the law of exponent,
(p^{m})^{n} = p^{mn}
Therefore, (5^{2})^{2} = 5 ^{2x2} = 5^{4} = 625
(iii) (5^{3}^{ }× 5^{4}) / 5^{2}
^{Ans: Using the law of exponents, we know that am × an = a(m+n)⇒ 53 × 54 = 5(3+4) = 57Now, we have (57) / 52Using the law of exponents, we know that am / an = a(mn)⇒ 57 / 52 = 5(72) = 55So, the simplified expression is 55. }
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1. How do you simplify expressions with exponents? 
2. What are the laws of exponents and how do they work? 
3. How do you convert a decimal number into standard form? 
4. How do you express large numbers in standard form? 
5. How can I practice working with exponents and powers effectively? 

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