Division of Exponents with the Same Base

πŸ†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:

aman=a(mβˆ’n)\frac {a^m}{a^n} =a^{(m-n)}

This property also concerns algebraic expressions.

Start practice

Test yourself on power quotient!

einstein

Choose the expression that is equal to the following:

\( a^5:a^4 \)

Practice more now

5 examples of division of powers with the same base

Exercise 1

5452=\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:

54βˆ’2=5^{4-2}=
52=5^2=

2525


Join Over 30,000 Students Excelling in Math!
Endless Practice, Expert Guidance - Elevate Your Math Skills Today
Test your knowledge

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.

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

Do not let the negative sign in the base confuse you.

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 βˆ’5-5 with a new exponent.

It will give us:

(βˆ’5)4=625 (-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.

12β‹…x2β‹…x44β‹…x3= \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 X among which is the sign to multiply.

We can add the exponents and obtain:

12β‹…x64β‹…x3= \frac{12\cdot x^{^6}}{4\cdot x^3}=

Now, we will realize that we can divide 12 12 by 4 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 X .

We will do it and obtain:

3β‹…x3 3\cdot x^3

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

3x3 3x^3


Do you know what the answer is?

Exercise 4

43β‹…28β‹…452β‹…4223= \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 4 ) and, between them, the multiplication sign.

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

48β‹…282β‹…4223= \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 2 ) and, between them, the multiplication sign. We can add the exponents and obtain:

48β‹…2824β‹…42= \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 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 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 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 2 and obtain:

46β‹…24= 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 4 into the natural number 2 2 .

We will express the 4 4 as 22 2^2 and obtain:

(22)6β‹…24= (2^2)^6\cdot2^4=

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

212β‹…24= 2^{12}\cdot2^4=

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

216=65,536 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

1X7X6= \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

x7x6=x(7βˆ’6)=x1=x \frac{x^7}{x^6}=x^{(7-6)}=x^1=x

Now, we are left with the fraction

1x \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:

1x=xβˆ’1 \frac{1}{x}=x^{^{-1}}


If you are interested in this article, you may also be interested in the following articles:

Fractions

How are fractions reduced?

Division and fraction line

The distributive property in the case of divisions

Division of whole numbers between parentheses in which there is a multiplication

Division of whole numbers between parentheses in which there is a division

In the Tutorela blog, you will find a variety of articles about mathematics.


Check your understanding

Division of Powers with the Same Base and Different Exponents Exercises

Exercise 1

Solve the following exercise:

2423= \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: 4βˆ’3=1 4-3=1

Answer

21=2 2^1=2


Exercise 2

8132= \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 34 3^4 .

Answer:

3432=32 \frac{3^4}{3^2}=3^2


Do you think you will be able to solve it?

Exercise 3

(4Γ—9Γ—11)a (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 4a9a11a 4^a9^a11^a

Answer:

4a9a11a 4^a9^a11^a


Exercise 4

(x2Γ—3)2= (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:

32=9 3^2=9

22=4 2^2=4

Answer:

9x4 9x^4


Test your knowledge

Exercise 5

(4Γ—7Γ—3)2= (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 -

42=16 4^2=16

72=49 7^2=49

32=9 3^2=9

Answer:

16Γ—49Γ—9=7056 16\times49\times9=7056


Exercise 6

(2Γ—3Γ—7Γ—9)ab+3 (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.

Answer:

2ab+33ab+37ab+39ab+3 2^{ab+3}3^{ab+3}7^{ab+3}9^{ab+3}


Do you know what the answer is?

Exercise 7

(5Γ—XΓ—3)3= \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:

(aΓ—bΓ—c)n=5nΓ—bnΓ—cn \left(a\times b\times c\right)^n=5^n\times b^n\times c^n

We will put the numbers in the formula.

(5Γ—XΓ—3)3=53Γ—X3Γ—33 \left(5\times X\times3\right)^3=5^3\times X^3\times3^3

Answer:

53Γ—X3Γ—33 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 X by itself X X times.


Check your understanding
Start practice