I’m going to explain how to operate with powers with the same and different base. You will learn how to multiply and divide powers of different base, both of variables and with numbers.

Índice de Contenidos

- 1 Multiplication of powers with the same base
- 2 Division of powers with the same base
- 3 Multiplications and divisions with powers with the same base
- 4 Multiplications and divisions of powers with different base
- 5 Operations with powers of numbers with different base
- 6 Operations with high powers at other powers
- 7 Operations with powers from different base elevated to other powers

## Multiplication of powers with the same base

When we have two powers multiplying, it is not a question of applying the property of the multiplication of powers with the same base and that’s it, but it is necessary to finish simplifying the operation with other properties.

Let’s see it with an example:

The first step is to check if they have the same base, that they have it.

Therefore, when we have multiplications with the same base, the power multiplication property is applied with the same base:

Keep the base and add the exponents.

In this case, we have a negative exponent, but it doesn’t matter, because we add a negative number and that’s it:

We are left with a negative base power (the exponent affects the minus sign because it is enclosed in parentheses), elevated to a negative exponent.

The next step is to apply the negative exponent property:

We pass that exponent to positive and then resolve the power in the denominator, which is negative because the exponent is odd:

As you can see we have applied two properties until we have simplified the operation. After adding or subtracting the exponents, always pass the exponent to positive.

## Division of powers with the same base

With the division of powers with the same base, the same thing happens as with multiplication. It is not enough to apply only the property of power division with the same base.

For example:

We have two powers that are dividing and have the same base, therefore, the first thing we must do is apply the property of the division of powers with the same base:

We maintain the base and subtract the exponents:

Nos has left a power with negative exponent, which we have to pass to positive exponent with this property:

That’s why we pass the power to the denominator with the positive exponent:

Summarizing, when we have multiplications or divisions of powers with the same base, we add or subtract exponents, that can be positive or negative and then we pass the exponent to positive.

## Multiplications and divisions with powers with the same base

In the same operation, we can have multiplications and divisions of powers with the same base. In other words, we would have a fraction with more than one power

In this case, we must apply the multiplication property, separately, in the numerator and in the denominator, then apply the division property and finally, pass the exponent to positive, if we have been negative.

Let’s take a slower look at an example:

We have an operation where several powers with the same base are multiplying and dividing.

We apply the multiplication property to the numerator and denominator. We maintain the base and add the exponents:

We are left with a fraction that has 2 particularities:

1 – We get a 2 raised to 0 in the numerator and we already know from the first property that any number raised to 0 is 1:

2 – We have a negative exponent in the denominator. We convert the exponent to positive by passing the power to the numerator. It is the same property as that of a power with negative exponent:

Continuing with our operation, we have the following:

Once we have passed the exponent to positive, the power can be resolved.

## Multiplications and divisions of powers with different base

In one operation we can find powers of different base, which are multiplying and dividing. Keep in mind that we can only multiply and divide powers when they have the same base.

If we have a multiplication of two powers that have different bases, such as this one:

We cannot operate with them because we cannot apply any property of the powers. It would stay as it is.

Therefore, the first thing we have to do is to look for the powers that have the same base, to multiply or divide them separately.

Let’s look at this concept with another example:

We have two bases: *x* e *y*.

With base x, we have two powers that are multiplying, so we can add the exponents. With base y, we can’t do anything and it stays as it is:

Do you see what the procedure is? You always have to look for powers of the same base to be able to apply the properties of the corresponding powers.

Let’s see another example:

We again have two bases: *x* and *y*.

We cannot multiply powers in the numerator and denominator, since we have powers of different base.

On the other hand, we do have power divisions with base x and with base y.

We divide separately with each of the bases. We treat them as if they were two fractions that are multiplying:

On the one hand, for the base *x*, the exponents are subtracted and on the other hand for the base *y* the exponents are also subtracted:

For each of the bases, we have a negative exponent left, which we turn positive by passing the power to the denominator:

Let’s see another example where we also have numbers, besides variables:

In this case we have on the one hand a fraction of numbers, on the other hand a division of powers with base x and on the other hand a division of base powers y.

With numbers we simplify the fraction, whose result is an integer:

With the bases x and y, we maintain the base and subtract the exponents. So we have our equation:

In the base y, we have the exponent equal to 0. We know by its corresponding property, that any variable or number a raised to 0 is 1, so we have:

And this would simplify the expression.

## Operations with powers of numbers with different base

When we work only with numbers and we have powers of different bases, we must look for the powers to have the same base, that is to say, we must express all the powers with the same base or if it is not possible to express all the powers with a single base, with the minimum possible number of bases.

And how do we express the number in another base? Then by breaking the number down into factors.

Let’s look at it with a very simple example:

In this multiplication of powers, in principle we can’t do anything, because we have a multiplication of powers of different base and we can’t add their exponents.

But we can decompose the 4:

Therefore, in the operation we are solving, we substitute 4 with its decomposition and in this way we have a multiplication of powers with the same base:

Before multiplying the powers, it is necessary to solve the parenthesis, multiplying the exponents:

Now we can multiply. We maintain the base and add the exponents

At the end, we can also solve the power.

Let’s see another example:

In principle, we have four bases: 2, 3, 4 and 9.

We want all powers to have the same base or the minimum number of bases possible. To do this, we must break down into prime factors the numbers that can be expressed in this way in the equation.

In this case we can break down 4 and 9, which we indicate in the equation as 2² and 3²:

We are left with two bases: 2 and 3.

The next step is to remove parentheses, multiplying the outer exponents by the inner exponents:

In the numerator we have two powers with base 2 multiplied, so we keep the base and add the exponents. We do the same in the denominator with two base powers 3:

Nos has remained a division of powers of base 2 and another of base 3. For each one we maintain the base and subtract the exponents:

Y with this we have finished simplifying the expression, since we do not have any negative exponent.

## Operations with high powers at other powers

Let us now see the steps to follow when we have multiplications or divisions with powers, which in turn are elevated to another power, such as:

We begin by multiplying the powers within the parenthesis:

Nos has been raised to another power. So now we multiply exponents:

We have made the negative exponent positive by passing it to the denominator.

We continue with a high power division at a negative exponent:

We begin operand within the parenthesis, subtracting the exponents:

We are left with one power raised to another power, so we multiply the exponents:

Let’s see a last example, in which we have all the operations with powers that we have seen until now:

First, we apply the property of power multiplication in the numerator and denominator. We maintain the base and add the exponents:

We are left with a division of powers. We maintain the base and subtract the exponents:

We are left with one power raised to another. Maintain the base and multiply the exponents:

At the end we have a power with negative exponent, which we turn positive by passing it to the denominator. Once we have the positive exponent, we can solve the power:

## Operations with powers from different base elevated to other powers

We are going to see the steps to follow when you have to simplify an operation in which you have multiplications and divisions of different base, which are also part of another power, as for example:

In the first place we simplify as much as possible inside the parenthesis.

Same as before, on the one hand we simplify the numbers and on the other hand, with each base x and y, we maintain the bases and subtract the exponents:

We can no longer operate within the parenthesis, so we proceed to resolve the parenthesis.

To solve the parenthesis, you have to multiply the exponent from outside by each of the exponents inside, according to this property:

Multiplying exponents leaves us:

Finally, we have to express the solution with all positive exponents.

We have negative exponents in the numerator and denominator.

I remind you that the powers with negative exponent that are in the numerator pass to the denominator with positive exponent and vice versa, according to this property:

Applied to our equation we have:

We finish the operation by solving the base power 2.