Exponents

“A repeated multiplication” is the first expression we must keep in mind when dealing with exponents, while “Power” is the term you will mostly use. “3 to the power of 5”, “7 square or 7 to the power of 2”, “9 to the power of 3”, “4 to the 5^th power”, etc. is how exponents are expressed. The best way to really understand what an exponent represents is by looking at some examples:

Let’s take 5 to the power of 3. As soon as we see that we are dealing with exponents, we must immediately think of “repeated multiplication”. To make things simpler, consider 5 as the base, while 3 tells us that the base must be multiplied by itself 3 times. In this case, the result is 5 * 5 * 5 which equals 125. Please do not be confused and multiply the base with its exponent because that is not exponents’ function.

Nonetheless, you must bear in mind that exponents have their rules by which they get multiplied, divided, and so on, and you have to learn them in order to avoid any mistake.

Let’s begin:

1. 0 to the power of any number will always remain 0. Why? Because any number that is multiplied by 0 will always give us 0. When 0 is multiplied by itself, obviously will give us 0.
2. 1 to the power of any number will always remain 1. Why? Because no matter how many times you multiply 1 by itself will always give us 1.
3. (2/5)². In such cases, the exponent 2 is valid for both the numerator and denominator. In this case, the result is 4/25. However, if we have 2²/5 then the exponent 2 is valid only for the numerator. Hence, the result is 4/5. The same applies for the denominator.
4. 10⁵. As expected, we immediately know that 10 is multiplied by itself for 5 times. However, the question is, “what is another way of expressing this exponent?” We know that 10 is the product of 2 * 5, therefore we can also write (2 * 5)⁵. In this case, the exponent 5 is valid for both 2 and 5, meaning that 10⁵ = 2⁵ * 5⁵.
5. (-5)². As soon as you see a negative base (such as -1, -7, -159, etc.) to an even power (such as 2, 4, 6, 8, 13568, etc.) you must know that the result will always be positive. Why? Because negative * negative always gives us positive. However, this is true only when you have the base in parentheses. This is because the power is valid for both the base and the sign in front of the base. Hence, when you see -5⁵ do not forget that the result will always remain negative because the power is valid only for the base and not for the sign in front of the base.
6. (-9)⁵. As soon as you see a negative base (such as the one in our example) to an odd power (such as 3, 5, 12467, etc.) do not forget that the result will always remain negative, because negative * negative * negative always gives us negative. Now let’s see -9⁵. When power is odd, (-9)⁵ = -9⁵. This is because the power is valid only for the base and not the sign in front of the base, meaning that the sign does not change while the base multiplies itself five times.
7. z⁴ * z⁹ = z⁴⁺⁹ = z¹³. What we realize from this multiplication is that when we multiply exponents with the same bases we leave the base the same while we add the exponents.
8. x¹³/x⁷=x^13-7=x⁶. When we have division (or fractions) and exponents share the same base, then we leave the base the same while we subtract exponents.
9. a⁵/a⁵=a⁰=1. We just followed the same rules as we did when dividing exponents with the same base. But, what we realized here is that any number to the power of 0 will always give us 1.
10. 1/3^-3 = 1/1/3³ = 3³/1 = 9. In this case, the exponent is valid only for the denominator. Since the exponent is negative, 3^-3 is the same as 1/3³. That’s why we have 1/1/9. Now that we are dealing with fractions, we can multiply outside fractions (1*9) with each other and inside fractions with each other (1*1) (following fractions rules) which will give us 9/1 which is equal to 9.
11. (b³)⁵ = b³*⁵ = b¹⁵. All we do in such cases is multiply the exponents, while leaving the base the same. As simple as that.

These are the main rules you must know about exponents. They are very simple and easy-to-learn. All you must do is study smart and hard, just like with everything else. Good luck.