## What is an Exponent?

### Exponents

The repeated addition of numbers can be written in short form (product form).

**Examples: **

Also, we can write the repeated multiplication of numbers in a short form known as exponential form.

For example, when 5 is multiplied by itself for two times, we write the product 5 × 5 in exponential form as 5^{2} which is read as 5 raised to the power two.

Similarly, if we multiply 5 by itself for 6 times, the product 5 × 5 × 5 × 5 × 5 × 5 is written in exponential form as 5^{6} which is read as 5 raised to the power 6.

In 5^{6}, the number 5 is called the base of 5^{6} and 6 is called the **exponent of the base**.

In general, we write,

An exponential number as b^{a}, where b is the base and a is the exponent.

The notation of writing the multiplication of a number by itself several times is called the **exponential notation** or **power notation**.

Thus, in general we find that :

If ‘a’ is a rational number then ‘n’ times the product of ‘a’ by itself is given as a × a × a × a ….. , n times and is denoted by a^{n}, where ‘a’ is called the base and n is called the exponent of a^{n}.

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**Examples**

1. Write the following statements as repeated multiplication and complete the table:

S.No. |
Statements |
Repeated Multiplication |
Short form |

(i) | 3 multiplied by 3 for 6 times | 3 × 3 × 3 × 3 × 3 × 3 = 729 | 3^{6} |

(ii) | 2 multiplied by 2 for 3 times | 2 × 2 × 2 | 2^{3} |

(iii) | 1 multiplied by 1 for 7 times | 1 × 1 × 1 × 1 × 1 × 1 × 1 | 1^{7} |

2. Write the base and exponent of following numbers. And also write in expanded form:

S.No. |
Numbers |
Base |
Exponent |
Expanded Form |
Value |

(i) | 3^{4} |
3 | 4 | 3 × 3 × 3 × 3 | 81 |

(ii) | 2^{5} |
2 | 5 | 2 × 2 × 2 × 2 × 2 | 32 |

(iii) | 3^{3} |
3 | 3 | 3 × 3 × 3 | 27 |

(iv) | 2^{2} |
2 | 2 | 2 × 2 | 4 |

(v) | 1^{7} |
1 | 7 | 1 × 1 × 1 × 1 × 1 × 1 × 1 | 1 |

### Exponents of Negative Integers

When the exponent of a negative integer is odd, the resultant is a negative number, and when the power of a negative number is even, the resultant is a positive number.When the exponent of a negative integer is odd, the resultant is a negative number, and when the power of a negative number is even, the resultant is a positive number.

or (a negative integer) an odd number = a negative integer.

(a negative integer) an even number = a positive integer.

**Examples: **

**Ex.1 **Express 144 in the powers of prime factors.

**Solution:**

144 = 16 × 9 = 2 × 2 × 2 × 2 × 3 × 3

Here 2 is multiplied four times and 3 is multiplied 2 times to get 144.

∴ 144 = 2^{4} × 3^{2}

**Ex.2 **Which one is greater : 3^{5} or 5^{3} ?

**Solution:
**3

^{5}= 3 × 3 × 3 × 3 × 3 = 9 × 9 × 3

= 81 × 3 = 243

and 5

^{3}= 5 × 5 × 5 = 25 × 5 = 125

Clearly, 243 > 125

∴ 3

^{5}> 5

^{3}

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