# Division of Exponents with the Same Base

๐Practice power quotient

## Division of Exponents with the Same Base

When we encounter exercises or expressions with terms that have the same base and between them the sign of division or fraction line, we can subtract the exponents.
We will subtract the exponent in the denominator from the exponent in the numerator.
That is:
"exponent of the denominator - exponent of the numerator" = new exponent
The result obtained from the subtraction is the new exponent and we will apply it to the original base.

Formula of the property:

$\frac {a^m}{a^n} =a^{(m-n)}$

This property also concerns algebraic expressions.

## Test yourself on power quotient!

$$\frac{2^4}{2^3}=$$

## 5 examples of division of powers with the same base

### Exercise 1

$\frac{5^4}{5^2}=$
Since the bases are the same we can subtract the exponents.
We will subtract the exponent in the denominator from the exponent in the numerator
and we will apply the new exponent (result of the subtraction) to the base:

$5^{4-2}=$
$5^2=$

$25$

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### Exercise 2

When we have a division of exponents with the same base, remember that we must subtract the exponent of the numerator from the exponent of the denominator.

$\frac{(-5)^6}{(-5)^2}=$

Simply subtract the exponents as indicated by the property we just studied: Exponent of the numerator minus the exponent of the denominator, thus you will get the base $-5$ with a new exponent.

It will give us:

$(-5)^{4}=625$

### Exercise 3

Let's move on to a slightly more complicated example in which we will also need to use the property of multiplication of powers with the same base.

$\frac{12\cdot x^{^2}\cdot x^4}{4\cdot x^3}=$

Don't panic. We will simply proceed according to the properties we have learned.

First, we will pay attention to the numerator. We will realize that there are equal bases $X$ among which is the sign to multiply.

We can add the exponents and obtain:

$\frac{12\cdot x^{^6}}{4\cdot x^3}=$

Now, we will realize that we can divide $12$ by $4$. In addition, we will use the property of Division of powers with the same base and subtract the exponent of the numerator from the exponent of the denominator since we have the same base $X$.

We will do it and obtain:

$3\cdot x^3$

We will multiply the $X$ by its coefficient and we will get:

$3x^3$

Do you know what the answer is?

### Exercise 4

$\frac{4^3\cdot2^8\cdot4^5}{2\cdot4^22^3}=$

Without the help of a calculator, we can quickly unravel the exercise and arrive at the correct result.

We'll start by looking at the numerator. We see that we have two equal bases ($4$) and, between them, the multiplication sign.

Therefore, we can add the exponents of the same base and obtain:

$\frac{4^8\cdot2^8}{2\cdot4^22^3}=$

Now let's look at the denominator. We will notice that there are equal bases ($2$) and, between them, the multiplication sign. We can add the exponents and obtain:

$\frac{4^8\cdot2^8}{2^4\cdot4^2}=$

Remember that, when there is no exponent, it means that the base is raised to the power of $1$ and we will not forget to add it.

Finally, we can take advantage of the property that deals with the division of powers of the same base.

We see that we have base $4$ both in the numerator and in the denominator. Therefore, we can subtract the exponent of the numerator from the exponent of the denominator of the same base.

In addition, the same happens with the base $2$, it exists in the numerator and in the denominator. We will subtract the exponent of the numerator from the exponent of the denominator of the base $2$ and obtain:

$4^6\cdot2^4=$

We got rid of the fraction and now we have a simple and delightful exercise.

Although you can solve it like this, you can use the properties of powers or laws of exponents again if you break down the base $4$ into the natural number $2$.

We will express the $4$ as $2^2$ and obtain:

$(2^2)^6\cdot2^4=$

Now we can multiply the power inside the parentheses by the one outside of them, which will give us:

$2^{12}\cdot2^4=$

Now we can add the exponents of the same base - 2 since between them there are multiplication signs, obtaining:

$2^{16}=65,536$

Recommendation:

If you have a division exercise where there is a base in the numerator and a different one in the denominator, try to perform some operation to equalize both bases, then you can proceed according to the property of division of powers of the same base.

### Exercise 5

$\frac{1}{\frac{X^7}{X^6}}=$

First, we will observe the fraction in the denominator of the exercise.

Here, two laws are used, first the law of the quotient of powers, according to which it is done

$\frac{x^7}{x^6}=x^{(7-6)}=x^1=x$

Now, we are left with the fraction

$\frac{1}{x}$

We know that this form can also be converted through the property of evaluating a negative power, so we can also write:

$\frac{1}{x}=x^{^{-1}}$

## Division of Powers with the Same Base and Different Exponents Exercises

### Exercise 1

Solve the following exercise:

$\frac{2^4}{2^3}=$

Solution:

According to the power rule, when there are two powers with the same base being divided by each other, the powers can be subtracted.

Therefore: $4-3=1$

$2^1=2$

### Exercise 2

$\frac{81}{3^2}=$

Solution:

According to the power law, when there are two powers with the same bases that are divided by each other, the powers can be subtracted. In this exercise, we must identify in the first step that the number \ (81 \) can be broken down into a power form, which is $3^4$.

$\frac{3^4}{3^2}=3^2$

Do you think you will be able to solve it?

### Exercise 3

$(4\times9\times11)^a$

Solution:

According to the power property, when we encounter an expression in which the power value appears throughout the product or in the entire exercise in which there are only multiplication operations among the members (using parentheses throughout the expression), we can take the power value and apply it to each product

That is, each of the products is powered.

Therefore $4^a9^a11^a$

$4^a9^a11^a$

### Exercise 4

$(x^2\times3)^2=$

Solution:

In this task, there is the use of two laws, both the multiplication of powers and the power of a power. Each of the products within parentheses receives the external power, since they have different bases and a multiplication operation between them. The power inside the parentheses is multiplied by the power outside of it, according to the law of a power of a power

Therefore:

$3^2=9$

$2^2=4$

$9x^4$

### Exercise 5

$(4\times7\times3)^2=$

Solution:

According to the power law, when we encounter an expression in which the power value appears throughout the product or in the entire exercise in which there are only multiplication operations between the products (using parentheses throughout the expression), we can take the value of the power and apply it to each product.

That is, each of the products is powered.

You can continue solving the exercise completely -

$4^2=16$

$7^2=49$

$3^2=9$

$16\times49\times9=7056$

### Exercise 6

$(2\times3\times7\times9)^{ab+3}$

Solution:

Even when the power coefficient is a compound exercise made up of several products, it does not modify the property. Each of the products within parentheses, as long as there is a multiplication operation between them, receives the power coefficient by itself.

$2^{ab+3}3^{ab+3}7^{ab+3}9^{ab+3}$

Do you know what the answer is?

### Exercise 7

$\left(5\times X\times3\right)^3=$

Solution:

It's important to remember that even when dealing with power properties, the order of arithmetic operations still exists. Therefore, it is possible (and correct) to double the products within parentheses before assigning them the power.

Using the formula:

$\left(a\times b\times c\right)^n=5^n\times b^n\times c^n$

We will put the numbers in the formula.

$\left(5\times X\times3\right)^3=5^3\times X^3\times3^3$

$5^3\times X^3\times3^3$

## Review Questions

How is the division of powers with the same base performed?

When we have a division of powers with the same base, we must subtract the exponent of the denominator from the exponent of the numerator. The result of this subtraction will be the new power and the base remains the same.

How to divide powers with the same base and different exponent?

We subtract the exponent of the denominator from the exponent of the numerator. The result of the subtraction is placed as the exponent of the common base.

What happens when the base and the exponent are the same?

It is a particular case of the power, in which we multiply the base $X$ by itself $X$ times.

## examples with solutions for power quotient

### Exercise #1

$\frac{81}{3^2}=$

### Step-by-Step Solution

First, we recognize that 81 is a power of the number 3, which means that:

$3^4=81$We replace in the problem:

$\frac{81}{3^2}=\frac{3^4}{3^2}$Keep in mind that the numerator and denominator of the fraction have terms with the same base, therefore we use the property of powers to divide between terms with the same base:

$\frac{b^m}{b^n}=b^{m-n}$We apply it in the problem:

$\frac{3^4}{3^2}=3^{4-2}=3^2$Therefore, the correct answer is option b.

$3^2$

### Exercise #2

$\frac{2^4}{2^3}=$

### Step-by-Step Solution

Let's keep in mind that the numerator and denominator of the fraction have terms with the same base, therefore we use the property of powers to divide between terms with the same base:

$\frac{b^m}{b^n}=b^{m-n}$We apply it in the problem:

$\frac{2^4}{2^3}=2^{4-3}=2^1$Remember that any number raised to the 1st power is equal to the number itself, meaning that:

$b^1=b$Therefore, in the problem we obtain:

$2^1=2$Therefore, the correct answer is option a.

$2$

### Exercise #3

Simplify the following expression:

$\frac{a^9}{a^x}$

### Step-by-Step Solution

In the question there is a fraction that has terms with identical bases in its numerator and denominator. Therefore, so we can use the distributive property of division to solve the exercise:

$\frac{c^m}{c^n}=c^{m-n}$

We apply the previously distributive property to the problem:

$\frac{a^9}{a^x}=a^{9-x}$

Therefore, the correct answer is (c).

$a^{9-x}$

### Exercise #4

Solve the following exercise

$\frac{a^{7y}}{a^{5x}}$

### Step-by-Step Solution

Let's consider that in the problem there is a fraction in the numerator and denominator with terms of identical bases, so we use the property of division between terms of identical bases to solve the exercise:

$\frac{c^m}{c^n}=c^{m-n}$We apply the previously mentioned property in the problem:

$\frac{a^{7y}}{a^{5x}}=a^{7y-5x}$Therefore, the correct answer is option A.

$a^{7y-5x}$

### Exercise #5

Simplify the following:

$\frac{a^5}{a^3}=$

### Step-by-Step Solution

Since a division operation between two terms with identical bases is required, we will use the power property to divide between terms with identical bases:

$\frac{c^m}{c^n}=c^{m-n}$Note that using this property is only possible when the division is carried out between terms with identical bases.

$\frac{a^5}{a^3}=a^{5-3}=a^2$Therefore, the correct answer is option A.
$a^2$