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What is an exponent; addition, subtraction, multiplication and division of exponents

exponents

Let’s say you have the number 2. If you multiply it by 2, you get 4. And if you multiply it by 2 again, you get 8. And if you want to keep multiplying it by 2 for, let’s say… 10 times, this is what it looks like written down:
\ 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2.

At this point, this expression is starting to look really messy and it’s really easy to lose count. If you forget how many times you multiplied a number with itself, you’d have to count it over and over again. And that would be a big waste of time. Math loves simplicity – and it is always trying to make really complicated and long expressions as short and neat as possible. In order to achieve that, exponents come in handy.

An exponent is a number which tells us how many times a number has been multiplied by itself. Exponents are written as smaller numbers that are placed on top right spot next to your original number. That number that is being multiplied over and over again is called the base of the exponentiation.

base of exponentation

It is said that, by default, every number which has 0 as its exponent is equal to 1. This means that, no matter how large is the base, if their exponent is equal to 0, that number is always equal to 1.

base exponent zero

Also, every number that doesn’t have an exponent attached to it, actually has the number 1 as its exponent. The number 1 is the default exponent of every number and is not necessary to write it down, but it helps us to see and compare numbers in some tasks.

base exponent of one

Also, the number 1 to any power is always equal to one, which is logical. If you multiply 1 as many times as you like with 1, you’ll always get 1.

one to any power equals one

We already mentioned that exponents indicate how many times a number is being multiplied by itself.

meaning of a power

Using this notation, we don’t have to count how many times we multiplied a number by itself, and it is very useful if we want our numbers to be as brief and tidy as possible. And most numbers can be significantly simplified. For example, if you multiplied the number 2 by itself 20 times, you’d get the number 1048576 as a result – which can be written as: .

Negative exponents

Sometimes, you will also run into negative exponents. You deal with them by turning them into positive exponents of the reciprocal value of the base. Like this:

negative exponents

Same goes the other way around. If you have a fraction, and an unknown as your denominator, the denominator can become a numerator simply by changing the sign of exponent. In some cases, this will prove to be a very useful feature, especially when we start working with inverse numbers and functions.

 

Addition

Now that you learned one more numerical notation, didn’t you start wondering about mathematical operations? How would you add or subtract exponents? These things aren’t as easy.

With great mathematical simplifications in notations come great complications in calculating.

For example, you know that \ 2^2 = 4 and \ 2^3 = 8 and you know that \ 4 + 8 = 12. But for \ 2^2 + 2^3, the answer is not that obvious. This is where it gets a bit more complicated. You can not add numbers that have different exponents or different bases. There are some tricks you can use to reduce your task, but that is as far as you can go. We’ll get to that later.

You will usually confront tasks that have unknowns. Same rule applies to them. If their exponent is equal to 1, they will be equal to their base, if it is zero they will be equal to 1. Again, you can only add exponents that have the same base and the same exponent.

Of course, if you have more than one unknown, you group sorts of unknowns together.

Let’s remember a simple equation.

\ x + 2 + 3x = 1

since \ x = x^1 and \ 1 = x^0 we can write our equation like this:

\ x^1 + 2 * x^0 + 3 * x^1 = 1 * x^0

How did you normally solve it? You added variables with x separately, and separately variables without x.

The same will apply to larger exponents:

\ x^{12} + 2 * x^2 + 3 * x^{12} = ?

\ x^{12} + 3 * x^{12} + 2 * x^2 = 4 * x^{12} + 2 * x^2
 

Basically, we group the variables with the same exponents.

Example – add exponents:

\ 2 * x^3 + 3 * x + 0.5 * x^2 + x^1 + 2 * x^7 + 3 * x^3 = ?

\ 2x^7 + (2 * x^3 + 3 * x^3) + 0.5 x^2 + (3x + x) = 2x^7 + 5x^3 + 0.5 x^2 + 4x
 

Subtraction

The same rules that apply to adding exponents, apply to subtracting as well.
You can only subtract numbers that have unknowns with the same exponent.

Example – subtract exponents:

4x^{12} - 0.25 x^4 + 2x^2 - 3x^2 - 3x^{12} = ?

(4x^{12} - 3x^{12}) - 0.25x^4 + (2x^2 - 3x^2) = x^{12} - 0.25x^4 - x^2

 

Multiplication

There are only two basic rules for multiplying exponents.
First is when you have the same base, their indexes add together.

multiplication of exponents

For example if you have \ 2^2 * 2^3 = 2^{2 + 3} = 2^5
Similar, if you come up with a negative exponent you can either leave it like that, or transform it into reciprocal fraction.

multiplication of negative exponents

For example if you get \ 2^{-2} * 2^{-3} = 2^{- 2 - 3} = 2^{-5} = (\frac{1}{2})^5
Second rule is if you have different bases, but same indexes, bases multiply and the index remains the same.

second rule in multiplication of exponents

For example if you have \ 2^2 * 3^2 = (2 * 3)^2 = 6^2

If you are ever in doubt which rules you can apply, you can always use those smaller numbers that you can easily calculate step by step, and see if what you got is true.

Example 1:

2^2 * 4^2 = ?

You know that for you to multiply two exponents you need the same base or the same exponent. Here you have niether. So how would you solve this? First step is to always try to turn every number to it’s lowest base. In this example you can see that the number 4 can be written as 2^2.
When we put that in the form of lowest bases we have:

What now? As you know the square represents the number multiplied by itself so \ (2^2)^2 can be written as \ 2^2 * 2^2 = 2^(2 + 2) = 2^4

That will be valid in general:

third rule in multiplication of exponents

Let’s continue with our example:

\ 2^2 * 4^2= 2^2 * (2^2)^2 = 2^2 * 2^4 (and now we have the same base) = \ 2^(2+4) = 2^6
Example 2:

(\frac{2}{3})^2 * 0.2^2 = ?

(\frac{2}{3})^2 * 0.2^2 = (\frac{2}{3})^2 * (\frac{2}{10})^2 = (\frac{2}{3})^2 * (\frac{1}{5})^2 = (\frac{2}{3} * \frac{1}{5})^2 = (\frac{2}{15})^2

What if unknowns get in a way? Well everything is kind of similar, you can only multiply unknowns that have either the same base or the same exponent.

Example 3: \ (x^2 y^3)(x^5 y^4 )

Since it is pure multiplication you can simply erase those braces and multiply what you can. Group your unknowns into sorts and then you can multiply. In this example you have two sorts of unknowns, ‘x’ and ‘y’. You deal with them individually, and then, and only then if they have the same exponent in the end you can put them together. \ (a^n * b^n = (a * b)^n).

(x^2 * y^3)(x^5 * y^4) = x^2 * x^5 * y^3 * y^4 = x^7 * y^7 = (xy)^7

If you get some kind of real number beside your unknown, you leave him alone, unless there are more than one real number in your task, then you multiply individually your real numbers, and individually unknowns.

Example 4:

4x^4 * 5x^2 = 4 * 5 * x^6 = 20x^6

And just the same if you have negative real numbers (just be careful, and remember that two minuses produce a plus), and the same if you have fractions.

Division

There are only two basic rules for dividing exponents.
First is when you have the same base, their indexes subtract.

 

division of exponents

For example \ 2^2 : 2 =2^{2 - 1} = 2^1 = 2 which you can easily check because 4 : 2 = 2.

division of negative exponents

For example \ 2^{-2} : 2^{-1} = 2^{-2-(-1)} = 2^{-1} = \frac{1}{2}, again you can check it \frac{1}{4} : \frac{1}{2} = \frac{1}{4} * \frac{2}{1} = \frac{1}{2}

Second rule is if you have different bases, but same indexes, bases divide and the index remains the same.

division of exponents different bases same index

for example \ 2^2 : 3^2 = (2 : 3)^2 = (\frac{2}{3})^2

Example 1:

\frac{4^2}{4^3} + \frac{1}{2} = ?

\frac{4^2}{4^3} + \frac{1}{2} = 4^{2 - 3} + \frac{1}{2} = 4^{-1} + \frac{1}{2} = \frac{1}{4} + \frac{1}{2} = \frac{1 + 2}{4} = \frac{3}{4}

 

Example 2:

\frac{4^5}{4^{-2}} - 0.2 * 4 + \frac{1}{2} * 2^8 = ?

 

\frac{4^5}{4^{-2}} - 0.2 * 4 + \frac{1}{2} * 2^8 = 4^{5 - (-2)} - \frac{2}{10} * 4 + \frac{2^8}{2^1} = 4^{5 + 2} - \frac{1}{5} * 4 + 2^{8 - 1} = 4^7 - \frac{4}{5} + 2^7

And if you run into some unknowns, you deal with them instinctively. All the same rules apply to them.

Example 3:

\frac{18x^5y^6a^2}{6xy^2a^5} = ?

Again, you sort your unknowns, and deal with them individually, as well as real numbers.

\frac{18x^5y^6a^2}{6xy^2a^5} = 3x^{5 - 1}y^{6 - 2}a^{2 - 5} = 3x^4y^4a^{-3}

And if you want all your exponents to be positive:

\ 3x^4y^4a^{-3} = \frac{3x^4y^4}{a^3}

But here you have to be careful, you can only do this if you have purely division and multiplication.
If you have something like this:

\frac{x^2y^3 + x^5y}{xy}

you can’t simply subtract exponents.

What you can do, is divide this fraction into two smaller fractions and than use your rules for division:

\frac{x^2y^3 + x^5y}{xy} = \frac{x^2y^3}{xy} + \frac{x^5y}{xy} = xy^2 + x^4

 

Exponents worksheets

Properties of exponents

  Numeric expressions (312.6 KiB, 489 hits)

  Algebraic expressions (450.1 KiB, 500 hits)

Basics of exponents

  Scientific notation (166.4 KiB, 575 hits)

  Scientific notation - Write in standard notation (187.0 KiB, 480 hits)

Operations with exponents

  Multiplication (195.3 KiB, 490 hits)

  Division (197.0 KiB, 441 hits)

  Raised to a power (174.1 KiB, 505 hits)

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