**Exponent **is any **no. raised to the base** which can also be seen as the power of the number that is **how many times the number is multiplied by itself**. It is represented in the form **a ^{b}**

**where a is the base and b is the power**.

**For Example**,

**the number 2**

^{5 }= 2 ∗ 2 ∗ 2 ∗ 2 ∗ 2 i.e 2 multiplied 5 times to itself. Hence here

**Base = 2, Exponent = 5. This can be read as 2 raised to the power 5**.

Note 1:When there is no given exponent to the number, its power of the number is one i.e. it is one time of itself.

Example:The no. 5 has base 5 has power one +1.

Note 2:Powers are useful for expressing large quantities.

Example:56,400,000,000,000,000,000 can be expressed easily as 5.64 x 10^{18 }

**Powers with **Negative** Exponents**

If a number says, n has negative exponent b as its power then it is basically the reciprocal of power. i.e.

**n**^{-b} = 1/n^{b}

^{-b}= 1/n

^{b}

**Example: **

The number 2

^{-3 }has base 2 and a negative exponent 3 i.e. -3.2

^{-3 }= 1/2^{3 }= 1/(2 * 2* 2) = 1/8

**Expanding Rational Numbers with Exponents**

A rational number can be expanded and represented in terms of power.

**Example:**

45679.32can be represented as 40000 + 5000 + 600 + 70 + 9 + 0.3 + 0.02 = 4×10^{4 }+ 5×10^{3 }+ 6×10^{2 }+ 7×10^{1}+ 9×10^{0 }+ 3×10^{-1}+2×10^{-2}

**Laws of Exponents and Powers**

**1. Exponents with **the **same base**

- When two exponential numbers are multiplied with the same base, then the exponents are added:
**n**^{a}* n^{b }= n^{(a + b)} - When two exponential numbers are divided with the same base, then the exponents are subtracted:
**n**^{a }/ n^{b }= n^{(a – b) }

**Example:**

(2

^{3 }* 2^{2 })/2^{4 }= 2^{(3 + 2)}/2^{4 }= 2^{5 – 4 }= 2

**2. Power of Power**

When a base having a power of power say** (n ^{a})^{b }**then the powers are multiplied.

**i.e. n**

^{a*b}.**Example:**

(3

^{2})^{3 }= 3^{2*3 }= 3^{6 }

**3. Same Exponents but Different Bases**

- When two numbers with multiplied with the same exponents then their bases are multiplied.
**n**^{a }x m^{a }= (n*m)^{a } - When two numbers with divided with the same exponents then their bases are divided.
**n**^{a }/ m^{a}= (n/m)^{a}

**Comparison of Quantities using Exponents**

When we need to compare two large or small quantities, we convert them to their standard exponential form and divide them.

**Example: Comparing the masses of the earth and that of the sun?**

**Solution:**

Mass of the Earth = 5.972 x 10

^{24 }Mass of the Sun = 1.989 x 10

^{30}Mass of the Sun/Mass of the Earth = (1.989 x 10

^{30}) / (5.972 x 10^{24})= 0.33 x 10

^{6 }= 3.3 x 10^{5}So the mass of the sun is approximately 10

^{5 }times that of earth.

### Some More Examples

**Question 1: Find values of **

**(i) 7 ^{3}**

**(ii) 7**

^{-3 }**Solution:**

(i)7^{3 }= 7 * 7 * 7 = 49 * 7 = 343(ii)7^{-3}= 1/7^{3 }= 1/(7 * 7 * 7) =1/343

**Question 2: Express the following in exponent and powers?****(i) 34500****(ii) 1/25**

**Solution:**

(i)34500 = 345 x 100 = 345 x 10^{2 }^{ }= 3.45 x 10^{2 }x 10^{2}(dividing 345 by 100 by shifting two decimel places to left and at the sametime multiplying by 100 or 10)^{2}

= 3.45 x 10^{4}(total power = 2 + 2)(ii)1/25 = 1/(5 x 5) = 1/5^{2}= 5^{-2 }(negative exponent)