## Exponents

An exponent is nothing but a number which is placed to the right-hand side of a number and in particular, above the number. In the image shown above, the exponent is n and b is the base number. In other words, to simplify things, 3^{3} means you need to multiply the base 3 with the number of times in the exponent.

So, 3^{3} = 3 x 3 x 3 = 9.

Exponents make the process of multiplication and division much easier. Because instead of writing 6x6x6x6, you can simply write it as 6^{4}. There is also more to exponents, as if you have two exponents of the same base such as:

X^{M} X^{N}, then all you need to do is add the exponents together like:

X^{M+N}

Take for example:

3^{2} x 3^{4} = 3^{(2+4)} = 3^{6}

Remember that this rule of adding the exponents, when it has the same base, is applicable only for multiplication.

3^{2} + 3^{4} is not equal to 3^{(2+4)}

**Example 1 – Expand 3^5.**

3^5 = 3 * 3 * 3 * 3 * 3

**Example 2 – Simplify 5 * 5* 5 * 5**

5 * 5 * 5 * 5 = 5^4

**Example 3 – Simplify 3^4 * 3^6**

3^4 * 3^6 = 3^ (4+6) = 3^10

## Radicals

A radical is nothing but a number that comes with a radical symbol (√). But don’t mistake it as a square root symbol even though it can be used to find the square root of a number. But it can also be used to find the cube root, fourth root etc.

Take for example,

√25 = 5, simply because 5×5 is 25.

Remember if there is a subscript before the radical symbol (√), then it indicates how many times the number needs to be multiplied to equal the radicand. A radicand is the number that is contained inside the radical symbol.

So √(64) is nothing but 8×8 or 8^{2}, hence the answer is 8.

**Example 1 – Solve √`144**

We need to see what number is squared (or essentially multiplied by itself) that will give us 144.

10^2 = 100

11^2 = 121

12^2 = 144

The answer is 12, and 144 is a perfect square.

**Example 2 – Solve √36**

We need to see what number is squared (or essentially multiplied by itself) that will give us 36.

4^2 = 16

5^2 = 25

6^2 = 36

The answer is 6, and 36 is a perfect square.

**Example 3 – Solve √30**

We need to see what number is squared (or essentially multiplied by itself) that will give us 36.

4^2 = 16

5^2 = 25

6^2 = 36

7^2 = 49

It is shown that there is no whole number that, when squared, will get 30. That means that 30 is not a perfect square. However, that number is between 5 and 6.

https://socratic.org/questions/is-50-a-perfect-square

## Scientific Notation:

Scientific notation is a method used to represent large numbers easily. Take, for example, the following number that is commonly used in Physics and Chemistry as the Avogadro’s number:

602,200,000,000,000,000,000,000

It is quite hard to write this number and that is the reason scientific notations are used. This Avogadro’s number can be easily written as a scientific notation by 6.022 × 10^{23}. Very simple, right? But how did you arrive at this notation?

Well, in order to convert a number into a scientific notation, you add a decimal point after the first number. The next step is to count the number of places after the decimal point, hence the Avogadro’s number can be easily written as:

6.022 x 10^{23}.

Let’s look at some examples:

46600000 = 4.66 x 10^{7}

0.00053 = 5.3 x 10^{-4}

45,667,323.003 = 4.5667323003 x 10^{7}