Properties of exponents & negative exponents

I’ve decided to make my blog posts more focused. I want to contribute in some way–I want to be able to give in a way that benefits myself and the other person–and I think that posts that are more focused will make it easier for people to engage because I won’t be traveling from one subject to another and possibly introducing them to things that are not so interesting to them.

Without further ado:

Negative exponents

There are several exponent properties. Exponents are numbers written above other numbers, and which indicate how many times 1 should be multiplied/ divided by the base (the lower number).


Above, n is the exponent, and X is the base.

There are both positive and negative exponents. When the exponent (n) is positive, then 1 is multiplied by the base n times.

2= 1 * 2 * 2 * 2

2= 8

When the exponent (n) is negative, then 1 is divided by the base |n| times. In this scenario, 2 is the base and -3 is the exponent. Also remember the math rule that dividing x by m is the same thing as multiplying x by m’s reciprocal.

x / m = x / (m/1) = x * (1/m)

The reciprocal of a number is that number with its numerator and denominator flipped. An integer like 5, for example, can be written as 5/1, meaning it can be written as a fraction. To find the reciprocal of 5, or 5/1, flip the numerator and denominator. So the reciprocal of 5 or 5/1 is 1/5. The reciprocal of 27 or 27/1 is 1/27.

When dividing x by m, you can also multiply if that is more convenient–you can multiply x by the reciprocal of m, and you will get the same thing you would get if you had done the division.

Anyway, if you have a negative exponent, you divide 1 by the base |n| times.


So we have to divide 1 by 2 three times. Not -3 times, just 3 times, because -3 is n, and the absolute value (|n|) of -3 is 3.

To divide by 2 is the same thing as to multiply by its reciprocal. 2 is 2/1, and its reciprocal is thus 1/2. Dividing by 2 = multiplying by 1/2.

2-3 = 1 * 1/2 * 1/2 * 1/2

2-3 = 1/8 or 1/23


2= 8 and 2-3 = 1/8

Exponent properties

  • Suppose you have (3x)and you want to distribute the exponent. The 3 and the x are being multiplied, and you distribute the exponent to both. (3x)= 33 * x3, or 27x3.


  • If you are multiplying two bases with same or different exponents, and the bases are the same, then the product is that base with an exponent that is the sum of both exponents. Or: Xa * X= Xa+b

Think about it. If you have this for example: 23 * 24 = 23+4 = 27, why does the rule make sense? Well, what does 2even mean? It means 2 multiplied by itself 3 times. You can imagine three 2s multiplying each other.

2 * 2 * 2

And now you are multiplying those 2s by more 2s. By four 2s, in fact, multiplying each other.

2 * 2 * 2 * 2 * 2 * 2 * 2

Which is the same thing as 27. When you multiply the same bases with different exponents, like we just did, you are taking the first list of the base multiplied by itself, and multiplying it by the second list of the base multiplied by itself.

(Below, the first list (of the multiplied base) is bold, and the second list is not:)

23 * 24 = ?

2 * 2 * 2 * 2 * 2 * 2 * 2 = ?

This new, longer list of the base multiplied by itself a bunch of times can be written in a simpler way that is easier to handle; a way that is more compact and easier to carry and work with. You can write that long list as the base with an exponent, the exponent referring to the number of times the base appears in the list.

23 * 24 = ?

2 * 2 * 2 * 2 * 2 * 2 * 2 = 2?

Since the number of times the base appears in the long list equals the number of times the base appears in each of the short lists, the exponent of the base representing the long list equals the number of times the base appeared in each short list.


  • If you have a base and an exponent within parentheses, and there is an exponent outside the parentheses, you multiply the exponents together to get the new exponent of the same base. (Xa)= Xa*b

An example would be (23)2. In this case, you multiply the 3 and 2 together to get 26.

With (23)2, what is happening is exactly what seems to be happening: the 2is being multiplied by itself twice. Imagine this happening:

(23)= (2 * 2 * 2)=2 * 2 * 2 * 2 * 2 * 2 = 26


  • When you are dividing one base by the same base, but both have different exponents, you subtract the second base’s exponent from the first base’s exponent to get the answer exponent, which you place above the base to get the answer.

X/ X= Xa-b

For example:

2/ 2= 23-4 = 2-1

This answer, 2-1, can be written differently. Remember that when an exponent is positive, you multiply 1 by that base the exponent number of times. But when an exponent is negative, you divide 1 by that base the exponent’s absolute value number of times.

So 2-1 = 1 / (2/1)

Which you know is the same as:

2-1 = 1 * (1/2)

So 2-1 = 1/2.

Want to know for sure that this is true? Whip out a calculator or use Google’s free online one and type in 1/2 and press enter. The calculator says:

1/2 = .5

Now type in 2-1 and the calculator ought to say:

2-1 = .5

They equal the same thing, which means that 1/2 = 2-1 without a doubt.

Thank you for reading this, and I hope you enjoyed it!


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