Exponents

A square lot has an area of 10,000 square meters.
This number can be acquired from 10 x 10 x 10 x 10 or 10⁴.

In the expression 10⁴, 4 is an exponent or index number and 10 is the base number. The exponent gives us the idea how many times the base is used as a factor. In 10⁴, 10 is used four times. It is read as "10 to the fourth power" or "10 raised to 4".

Any number including 0 with exponent 0 has value 1. Any number with exponent 1 is the number itself.

Examples:
1. 5³ = 5 x 5 x 5 = 125
2. 13⁴ = 13 x 13 x 13 x 13 = 28 561
3. 2 x 2 x 2 x 2 x 2 x 2 = 2⁶
4. aaabbc = a³b²c  where a, b, and c are real numbers.
5. 21⁰ = 1
6. 10, 356¹ = 10,356

Laws of Exponents

There are laws to be followed when multiplying or dividing numbers (or letter variables) with exponents. But remember to keep the base numbers in the same value.

Law 1:  When multiplying, add the indices.

        a^x $\cdot$ a^y  =  a^{x + y}

        Example:   3² x 3⁷  = 3²⁺⁷  = 3⁹

Law 2:  When dividing, subtract the indices.

         $\frac{a^x}{a^y}= a^{x-y}$

        Example:   $\frac{2^10}{2^4}= 2^{10-4}=2^6$

Law 3: Any number raised to zero is equivalent to 1.

        $a^0=1$

        Example 1:  $\frac{5^2}{5^2}= 5^{2-2}=5^0=1$

        Example 2:  $100,000^{0}=1$

Law 4: When an index form is raised to another power, multiply the indices.

        $(a^x)^y= a^{x \cdot y}$

       

        Example 1: (2³)⁵ = 2^{3 • 5} = 2¹⁵

        Example 2:

Law 5: When multiplying or dividing numbers that is raised to a single power, distribute the power to the numbers.

        $(a \cdot b)^x= a^x\cdot b^x$

       

        Example 1: $(3 \cdot 4)^2= 3^2 \cdot 4^2$

        Example 2: $(\frac{1}{7})^2 = \frac{1^2}{7^2}$

How to find the exponent of a number?
Let's have the number 16 for example. We need to check for divisibility.
We need to think of a fundamental number where 16 is divisible from.

Now, 16 is divisible by 2, 4, and 8 but 2 is the most fundamental number among them so

16 ÷ 2 = 8
  8 ÷ 2 = 4
  4 ÷ 2 = 2
  2 ÷ 2 = 1  end
How many times did we divide 16 by 2 until we get 1?
It is four times! So we have 2⁴ = 2 x 2 x 2 x 2 = 16.

Another example, 108.
We divide by 2 since the number is divisible by 2.

108 ÷ 2 = 54
  54 ÷ 2 = 27  but 27 is divisible by 3.
  27 ÷ 3 = 9
    9 ÷ 3 = 3
    3 ÷ 3 = 1  end
And so the expression of 108 in exponent form is 108 = 2 x 2 x 3 x 3 x 3 =  2² 3³
This is like the 4th item in the Examples above.

It is also a way to find the factors of a number which will be discussed in details in prime factorization section.