How is the addition and subtraction of powers that have the same base made?
The properties of the powers are applied when we have multiplication or division of powers, but how to operate with the addition and subtraction of powers? Can the properties of the powers also be applied?
When you find yourself with addition and subtraction of powers, you should not be confused with adding or subtracting the exponents. Remember that the properties of the powers are not applicable with addition and subtraction, only with multiplications and divisions.
Next I’ll explain how to operate when you have addition and subtraction of power with the same base.
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Addition and subtraction of power from the same base, when the base is a variable
Let’s start by looking at what you have to do when you find a sum and subtraction of powers that are based on a variable, such as this sum:
The biggest mistake and what you NEVER have to do is add up the exponents:
You can’t add up the exponents. Only when you have a multiplication of powers with the same base.
So, how do you solve this sum of powers with the same base and exponent?
In this particular case, it adds up as follows:
We have two similar terms, that is, we add the number of times x² appears.
I remind you that similar terms are those that have the same variable elevated to the same exponent.
and what is to be done when the terms are not similar?
Let’s see another sum of powers, where the terms are not alike, like this one:
Nothing can be done in this sum. The exponents cannot be added because it is not a multiplication of powers nor can the terms be added because they are not similar. So, it stays as it is:
Therefore, when the base of the powers are variable and they are being added or subtracted between them, you have to look at if they are similar terms or not in order to add them.
Addition and subtraction of powers of equal base, when the base is a number
We continue with operations that have with sum and subtraction of powers but this time, when the bases are numbers, as for example this sum of powers with base 3:
As before, what you should NEVER do is keep the base and add up the exponents:
It is not a multiplication of powers and that property cannot be applied.
When you have numbers as a base, what you have to do is solve each power separately. Each term is an independent power.
If you have powers with the negative exponent, first you have to convert it into positive, passing the power to the denominator, if not, you will not be able to solve the power.
Continuing with our previous example, the 3 that is raised to -2, we have previously turned it into positive and the 3 elevated to the cube we have solved it directly:
Once the powers are solved, we’ll have to solve a sum of fractions:
Let’s look at another example very similar to the one we saw in previous lessons with multiplications and divisions of powers, but in this case, with addition and subtraction:
Isn’t it very tempting to add and subtract the exponents while maintaining the same base? Well, don’t even think about it, let alone do it.
As I have just told you, the correct way to operate is by resolving each power separately.
We have a 2 elevated to 0, which you know is equal to 1.
The powers with negative exponents go to the denominator to convert them into positive and once in the denominator you can solve:
We’ve got one fraction operation left, and we’re still working it out:
Therefore, when the powers are based on a number and are adding or subtracting, you have to solve each power separately and then finish the operation with numbers and fractions you have left.
Addition and subtract powers squared
Another very repeated case, where mistakes are made with the powers is when we have a sum or a subtraction squared , like this:
The tendency of the whole world in the beginning is to multiply the exponent from outside by each of the exponents within the parentheses.
But this is another thing you should NEVER do. In the case of a sum the result is not equal to multiplying the exponents:
So how does it work out?
The square of a sum is a special product and it is resolved as follows: