Exponents and Powers

Do you know what the mass of earth is? It is

5,970,000,000,000,000,000,000,000 kg!

Can you read this number?

Mass of Uranus is 86,800,000,000,000,000,000,000,000 kg.

Which has greater mass, Earth or Uranus?

Distance between Sun and Saturn is 1,433,500,000,000 m and distance between Saturn and Uranus is 1,439,000,000,000 m. Can you read these numbers? Which distance is less?

These very large numbers are difficult to read, understand and compare. To make these numbers easy to read, understand and compare, we use exponents. In this Chapter, we shall learn about exponents and also learn how to use them.

We can write large numbers in a shorter form using exponents.

Observe 10, 000 = 10 × 10 × 10 × 10 = 10^{4}

The short notation 104 stands for the product 10×10×10×10. Here ‘10’ is called the **base **and ‘4’ the **exponent**. **The number** **10 ^{4} is read as 10 raised to the power of 4 **or simply as fourth power of 10. 10

We can similarly express 1,000 as a power of 10. Since 1,000 is 10 multiplied by itself three times,

1000 = 10 × 10 × 10 = 10^{3}

Here again, 10^{3} is the exponential form of 1,000.

Similarly, 1,00,000 = 10 × 10 × 10 × 10 × 10 = 10^{5}

10^{5} is the exponential form of 1,00,000

In both these examples, the base is 10; in case of 103, the exponent is 3 and in case of 10^{5} the exponent is 5.

We have used numbers like 10, 100, 1000 etc., while writing numbers in an expanded form. For example, 47561 = 4 × 10000 + 7 × 1000 + 5 × 100 + 6 × 10 + 1

This can be written as 4 × 10^{4} + 7 ×10^{3 }+ 5 × 10^{2} + 6 × 10 + 1.

Try writing these numbers in the same way 172, 5642, 6374.

In all the above given examples, we have seen numbers whose base is 10. However the base can be any other number also. For example:

81 = 3 × 3 × 3 × 3 can be written as 81 = 3^{4}, here 3 is the base and 4 is the exponent.

Some powers have special names. For example,

10^{2}, which is 10 raised to the power 2, also read as ‘10 squared’ and

10^{3}, which is 10 raised to the power 3, also read as ‘10 cubed’.

Can you tell what 5^{3} (5 cubed) means?

5^{3} means 5 is to be multiplied by itself three times, i.e., 5^{3} = 5 × 5 × 5 = 125

So, we can say 125 is the third power of 5.

What is the exponent and the base in 5^{3}?

Similarly, 2^{5} = 2 × 2 × 2 × 2 × 2 = 32, which is the fifth power of 2.

In 2^{5}, 2 is the base and 5 is the exponent.

In the same way,

243 = 3 × 3 × 3 × 3 × 3 = 3^{5}

64 = 2 × 2 × 2 × 2 × 2 × 2 = 2^{6}

625 = 5 × 5 × 5 × 5 = 5^{4 }

Find five more such examples, where a number is expressed in exponential form. Also identify the base and the exponent in each case.

You can also extend this way of writing when the base is a negative integer.

What does (–2)^{3} mean?

It is (–2)^{3} = (–2) × (–2) × (–2) = – 8

Is (–2)^{4} = 16? Check it.

Instead of taking a fixed number let us take any integer a as the base, and write the numbers as,

a × a = a^{2} (read as ‘a squared’ or ‘a raised to the power 2’)

a × a × a = a^{3} (read as ‘a cubed’ or ‘a raised to the power 3’)

a × a × a × a = a^{4} (read as a raised to the power 4 or the 4^{th} power of a)

..............................

a × a × a × a × a × a × a = a^{7} (read as a raised to the power 7 or the 7^{th} power of a) and so on.

a × a × a × b × b can be expressed as a^{3}b^{2} (read as a cubed b squared)

a × a × b × b × b × b can be expressed as a^{2}b^{4} (read as a squared into b raised to the power of 4).

Express 256 as a power 2.

We have 256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2.

So we can say that 256 = 2^{8}

Which one is greater 2^{3} or 3^{2} ?

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}

Which one is greater 8^{2} or 2^{8}?

8^{2} = 8 × 8 = 64

2^{8} = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 256

Clearly, 2^{8} > 8^{2}

Expand a^{3} b^{2}, a^{2} b^{3}, b^{2} a^{3}, b^{3} a^{2}. Are they all same?

a^{3} b^{2} = a^{3} × b^{2}

= (a × a × a) × (b × b)

= a × a × a × b × b

a^{2} b^{3} = a^{2} × 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 }× a^{2}

= b × b × b × a × a

Note that in the case of 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.

Express the following numbers as a product of powers of prime factors:

(i) 72 (ii) 432 (iii) 1000 (iv) 16000

(i) 72 = 2 × 36 = 2 × 2 × 18

= 2 × 2 × 2 × 9

= 2 × 2 × 2 × 3 × 3 = 2^{3} × 3^{2}

Thus, 72 = 2^{3} × 3^{2} (required prime factor product form)

(ii) 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} (required form)

(iii) 1000 = 2 × 500 = 2 × 2 × 250 = 2 × 2 × 2 × 125

= 2 × 2 × 2 × 5 × 25 = 2 × 2 × 2 × 5 × 5 × 5

or 1000 = 2^{3} × 5^{3}

Atul wants to solve this example in another way:

1000 = 10 × 100 = 10 × 10 × 10

.= (2 × 5) × (2 × 5) × (2 × 5) (Since10 = 2 × 5)

= 2 × 5 × 2 × 5 × 2 × 5 = 2 × 2 × 2 × 5 × 5 × 5

or 1000 = 2^{3} × 5^{3}

Is Atul’s method correct?

(iv) 16,000 = 16 × 1000 = (2 × 2 × 2 × 2) ×1000 = 24 ×103 (as 16 = 2 × 2 × 2 × 2)

= (2 × 2 × 2 × 2) × (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}

Work out (1)^{5}, (–1)^{3}, (–1)^{4}, (–10)^{3}, (–5)^{4}.

(i) We have (1)^{5} = 1 × 1 × 1 × 1 × 1 = 1

In fact, you will realise that 1 raised to any power is 1.

(ii) (–1)^{3} = (–1) × (–1) × (–1) = 1 × (–1) = –1

(iii) (–1)^{4} = (–1) × (–1) × (–1) × (–1) = 1 ×1 = 1

You may check that (–1) raised to any odd power is (–1),

and (–1) raised to any even power is (+1).

(iv) (–10)^{3} = (–10) × (–10) × (–10) = 100 × (–10) = – 1000

(v) (–5)^{4 }= (–5) × (–5) × (–5) × (–5) = 25 × 25 = 625

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