# Negative Exponents

**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.Hey! Looking for some great resources suitable for young ones? You've come to the right place. Check out our

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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 decimal 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)