Chapter Notes: Exponents & Powers

# Exponents and Powers Class 7 Notes Maths Chapter 10

Do you know that the distance between the Sun and Earth is 1,49,60,00,00 kmThat'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: 1.496×108 kilometers.

But what does 10mean? Let's understand this along with the chapter.

## Exponents

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 83 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(10 raised to the power of 4).

104 is called the exponential form of 10,000.

Question for Chapter Notes: Exponents & Powers
Try yourself:The exponential form of 100000 is

### Exponents and Powers Formulas

If p is a rational number and has a non-zero value, m is a natural number, then, p × p × p × p ×…..× p(m times) is written as pm, where p is the base number and m is the exponent value and pm is the power and ‘pm’ is said as ‘p – raised to the power m’. This is the general representation of exponents and powers.
Example: 5 x 5 = 52, where 5 is the base number and 2 is the exponent.

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 = 28.

### General form

If a number is multiplied by itself m times, then it can be written as:
b x b x b x b x b...m times = b

• Here, b is called the base, and n is called the exponent, power, or index.
• Numbers raised to the power of two are called square numbers.

Example: Which one is greater 23 or 32

Ans: We have, 23 = 2 × 2 × 2 = 8 and 3 2 = 3 × 3 = 9.
Since 9 > 8, so, 32 is greater than 23.

Example: Expand a3b2, a2b3, b2a3, b3a2 . Are they all same?

Ans:  a3b2  = a3 × b2 = (a × a × a) × (b × b) = a × a × a × b × b

a2b3 = ax b3 = (a × a) × (b × b × b)

b2a= b2 × a3 = (b × b) × (a × a × a)

b3a2 = b3 x a2 = (b × b × b) × (a × a)

In the terms a3b2 and a2b3 the powers of a and b are different. Thus a3b2 and a2b3 are different. On the other hand, a3b2 and b2a3 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, a3b2 = a3× b2 = b2 × a3 = b2a3

Similarly, a2b3 and b3a2 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
= 24 × 33

(ii) 16000

Ans:16,000 = 16 × 1000 = (2 × 2 × 2 × 2) ×1000 = 24 ×103

(as 16 = 2 × 2 × 2 × 2)
= (2 × 2 × 2 × 2)  x (2 × 2 × 2 × 5 × 5 × 5)
= 24 × 23 × 53
(Since 1000 = 2 × 2 × 2 × 5 × 5 × 5)
= (2 × 2 × 2 × 2 × 2 × 2 × 2 ) × (5 × 5 × 5) or, 16,000
= 27 × 53

Question for Chapter Notes: Exponents & Powers
Try yourself:The exponential form of 64 is

### Some Important Points

• Numbers raised to the power of two are called square numbers. Square numbers are also read as two-square, three-square, four-square, five-square, and so on.
• Numbers raised to the power of three are called cube numbers. Cube numbers are also read as two-cube, three-cube, four-cube, five-cube, and so on.
• Negative numbers can also be written using exponents.
• If an = b, where a and b are integers and n is a natural number, then an is called the exponential form of b.
• The factors of a product can be expressed as the powers of the prime factors of 100.
• This form of expressing numbers using exponents is called the prime factor product form of exponents.
• Even if we interchange the order of the factors, the value remains the same.
• So a raised to the power of x multiplied by b raised to the power of y, is the same as b raised to the power of y multiplied by a raised to the power of x.
• The value of an exponential number with a negative base raised to the power of an even number is positive.
• If the base of two exponential numbers is the same, then the number with the greater exponent is greater than the number with the smaller exponent.
• A 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 known as its standard form.

## Laws of Exponents

1. Multiplying Powers with the Same Base: For any non-zero integer a and whole number m and n, am × an = am+n.
Example: let us multiply 22 × 23. Using the rule, 22 × 23 = 2 (2 + 3) = 25.
2.  Dividing Powers with the Same Base :
For any non-zero integer a and whole number m and n (m > n), am ÷ an = am-n
Example: 5/ 52 = = 5 × 5 × 5 × 5 = 56-2
3. Taking Power of a Power :
For any non-zero integer a and whole number m and n, (am)n = amn
Example: (32)4 = 32 × 32 × 32 × 32 = 3(2+2+2+2) = 38 = 32 × 4
(8 is the product of 2 and 4)
4.  Multiplying Powers with the Same Exponents :
For any non-zero integers a and b and whole number m, am × bm = (ab)m
Example: 23 × 33 = (2 × 2 × 2) × (3 × 3 × 3) = (2 × 3) × (2 × 3) × (2 × 3) = 6 × 6 × 6 = 63
In general, for any non-zero integer am × bm = (ab)m (where m is any whole number)
5. Dividing Powers with the Same Exponents :
For any non-zero integers a and b and whole number m, a÷ bm = am/bm = (a/b)m
Example:
We may generalize (am/bm) = where a and b are any non-zero integers and m is a whole number.

Question for Chapter Notes: Exponents & Powers
Try yourself:(22)3 =

### Decimal Number System

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)+( 8×10)+( 7×10)+( 2×10
= 4872

### Expressing Large Numbers in the Standard Form

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 × 1011m.

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) 23 x 22

Ans: Numbers raised to the power of three are called cube numbers From the law of exponent
we know, pm x pn =p(m+n)
Therefore, 23 x 22 = 2(3+2) = 25 = 2 x 2 x 2 x 2 x 2 = 32

(ii) (52)2

Ans: We know, by the law of exponent,
(pm)n = pmn
Therefore, (52)2 = 5 2x2 = 54 = 625

(iii) (53 × 54) / 52

Ans: Using the law of exponents, we know that am × an = a(m+n)
53 × 54 = 5(3+4) = 57
Now, we have (57) / 52
Using the law of exponents, we know that am / an = a(m-n)
⇒ 57 / 52 = 5(7-2) = 55
So, the simplified expression is 55.

Question for Chapter Notes: Exponents & Powers
Try yourself:If 23 × 24 = 2?, then ? =

The document Exponents and Powers Class 7 Notes Maths Chapter 10 is a part of the Class 7 Course Mathematics (Maths) Class 7.
All you need of Class 7 at this link: Class 7

## Mathematics (Maths) Class 7

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## FAQs on Exponents and Powers Class 7 Notes Maths Chapter 10

 1. How do you simplify expressions with exponents?
Ans. To simplify expressions with exponents, you can use the laws of exponents such as the product rule, quotient rule, power rule, and zero exponent rule. These rules help you manipulate and simplify expressions involving exponents.
 2. What are the laws of exponents and how do they work?
Ans. The laws of exponents are rules that govern how to manipulate expressions with exponents. These laws include the product rule (a^m * a^n = a^(m+n)), the quotient rule (a^m / a^n = a^(m-n)), the power rule ((a^m)^n = a^(m*n)), and the zero exponent rule (a^0 = 1). These laws help simplify and solve problems involving exponents.
 3. How do you convert a decimal number into standard form?
Ans. To convert a decimal number into standard form, you need to write the number in scientific notation. Write the decimal number as a number between 1 and 10 multiplied by a power of 10. For example, 0.0034 can be written as 3.4 x 10^-3 in standard form.
 4. How do you express large numbers in standard form?
Ans. To express large numbers in standard form, you need to write the number as a number between 1 and 10 multiplied by a power of 10. For example, 5,600,000 can be written as 5.6 x 10^6 in standard form.
 5. How can I practice working with exponents and powers effectively?
Ans. To practice working with exponents and powers effectively, you can solve various problems involving exponents, use online resources and practice worksheets, and review the laws of exponents regularly. Consistent practice and understanding of the rules will help you master working with exponents and powers.

## Mathematics (Maths) Class 7

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