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, 33 means you need to multiply the base 3 with the number of times in the exponent.

So, 33 = 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 64. There is also more to exponents, as if you have two exponents of the same base such as:

XM XN, then all you need to do is add the exponents together like:

XM+N

Take for example:
32 x 34 = 3(2+4) = 36

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

32 + 34 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

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 82, 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 × 1023. 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 1023.

Let’s look at some examples:

46600000 = 4.66 x 107
0.00053 = 5.3 x 10-4
45,667,323.003 = 4.5667323003 x 107

https://calcworkshop.com/exponents/scientific-notation/