Exponents And Powers

Exponents Meaning

Exponents are the repeated multiplication of any number. The number 1,000,000 can be written as

1,000,000 = 10 × 10 × 10 × 10 × 10 × 10 = 106, where 6 is the power of 10.

Thus, when 10 is multiplied 6 times we get 1,000,000. It can be expressed using power as 106. 10 is the base and 6 is the exponent. It is read as ’10 raised to the power of 6’. Exponents and Powers class 8 is the most important topic, where this concept is explained in detail.

Laws of Exponents

  • Bases – multiplying the like ones – add the exponents and keep base same. (Multiplication Law)
  • Bases – raise it with power to another – multiply the exponents and keep base same.
  • Bases – dividing the like ones – ‘Numerator Exponent – Denominator Exponent’ and keep base same. (Division Law)

Let ‘a’ is any number and ‘m’ ,  ‘n’ are positive integers, then

  1. Multiplication Law

           am × an  = am+n

  1. Division Law

          am \(\div\) an  = am / an  = am-n

  1. Negative exponent 

            a-m  = 1/am 

Exponent Rules

i) a0 = 1

ii) (am)n = a(mn)

iii) am × bm =(ab)m

iv) am/bm = (a/b)m

Exponents and Power Questions

Example 1: Write 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 in exponent form.

Solution:

In this problem 7s are written 8 times, so the problem can be rewritten as an exponent of 8.

7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 = 78.

Example 2: Write below problems like exponents

  1. 3 x 3 x 3 x 3 x 3 x 3
  2. 7 x 7 x 7 x 7 x 7
  3. 10 x 10 x 10 x 10 x 10 x 10 x 10

Solution: 

  1. 3 x 3 x 3 x 3 x 3 x 3 = 36
  2. 7 x 7 x 7 x 7 x 7 = 75
  3. 10 x 10 x 10 x 10 x 10 x 10 x 10 = 107 

Example 3: Simplify 253/53 

Solution: 

 Using Law: am/bm = (a/b)m

253/53  can be written as (25/5)3  = 53 = 125.

Using Powers of Ten as Exponents

Scientific notation uses power of ten expressed as exponents, so we need a little background before we can jump in. In this concept, we round out your knowledge of exponents, which we studied in previous classes.

The distance between the Sun and the Earth is 149,600,000 kilometers. The mass of the Sun is 1,989,000,000,000,000,000,000,000,000,000 kilograms. The age of the Earth is 4,550,000,000 years. These numbers are way too large or small to memorize in this way.  With the help of exponents and powers these huge numbers can be reduced to a very compact form and can be easily expressed in powers of 10.

Now, coming back to the examples we mentioned above, we can express the distance between the Sun and the Earth with the help of exponents and powers as following:

Distance between the Sun and the Earth 149,600,000 = 1.496× 10 × 10 × 10 × 10 × 10× 10 × 10 = 1.496× 108 kilometers.

Mass of the Sun: 1,989,000,000,000,000,000,000,000,000,000 kilograms = 1.989 × 1030 kilograms.

Age of the Earth:  4,550,000,000 years = 4. 55× 109 years

Define Powers and Exponents

The exponent is a simple but powerful tool. It tells us how many times a number should be multiplied by itself to get the desired result. Thus any number ‘a’ raised to power ‘n’ can be expressed as:

Exponents And Powers

Here a is any number and n is a natural number.

an is also called the nth power of a.

‘a’ is the base and ‘n’ is the exponent or index or power.

 ‘a’ is multiplied ‘n’ times thereby exponentiation is the shorthand method of repeated multiplication.

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This section covers the few important concepts related powers and exponents. For any further query on this topic install BYJU’S the learning app and excel your knowledge.

Practise This Question

The value of  27 is _____.

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