Exponents or Exponential form is an easy way of expressing a same number multiplied to itself one or more times.

The exponential form of 3 × 3 is 3^{2}.

3^{2} is the short form for “3 multiplied by itself 2 times”.

2 × 2 × 2 is shortened as 2^{3}.

2^{3} is exponential expression signifying that 2 is multiplied 3 times to itself.

In 3^{2}, 3 is base and 2 is exponent

Base denotes the number that is multiplied to itself, while Exponent, number of times that the number is so multiplied.

The product

a × a × a × a × ……. n times is a^{n}

In a^{n}, a is base and n is exponent.

Note:

a can be any number, but n is a natural number (the numbers that we use for counting)

**a to the power 2** is called a squared.

For example,

3 to the power 2 written as 32 is called 3 squared.

**a to the power 3 **is called a cubed.

For example,

3 to the power 3 written as 33 is also called 3 cubed.

Now, if you wish to set off with your lesson on Exponents, then click on any of the links below:

Exponents

Negative Sign of Base

Laws of Exponents

nth root of a negative number:

Or, if you wish to capture a terse overview of each Exponents Formula, then go through each of the following header-links.

You can also click the header-links to take you to the page discussing in detail the specific Exponents concept:

Laws of Exponents:

3. Let a be not equal to 0. Then a^{0} = 1.

4. the negative sign of the base:

Remains if power is odd integer

Example: (-2)^{3} = - 8

Disappears, if power is even integer

Example: (-2)^{4} = 16

**2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 **

Is there an easier, shorter-form of writing or expressing the above long product?

The answer is an emphatic and relieving Yes.

It is **2 ^{8}**, read as

**2 ^{8} ** is called the exponential form of 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

In 2^{8}, 2 is called base and 8 is called exponent.

**Note: **

1. 5^{2} is 5 to the power 2, but a special name is **“5 squared”**

2. 100^{3} is 100 to the power 3, but a special name is **“100 cubed”**

So, we see that there are special names in calling the exponential forms when the exponents are 2 and 3.

64 = 2 × 2 × 2 × 2 × 2 × 2 = 2

-1/128 = (-1/2) × (-1/2) × (-1/2) × (-1/2) × (-1/2) × (-1/2) × (-1/2) = (-1/2)

“Minus one-by-two to the power 7”

**Note:**

1. - 8 = -2 × -2 × -2 = (-2)^{3} = - 8

2. 16 = 2 × 2 × 2 × 2 OR

-2 × -2 × -2 × -2 = (-2)^{4} = 16

From the above two examples, we note the following two very important rules:

**If the exponent is an odd integer, the negative sign of the base inside the brackets will not go away;
And if the exponent is an even integer, the negative sign of the base inside the brackets will go away. **

Let us summarize the above two points in blue in the table below:

**Very Important Point:**

Any number to the power 0 is equal to 1, i.e.

a^{0} = 1, a is any number, except 0.

Negative Integral Exponents:

If a is any number and n is a __positive integer__, then

a ^{–n} = 1/a^{n}

For example,

2^{–2} = 1/2^{2} = ¼

4^{ –3} = 1/4^{3} = 1/64

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