Dividing Exponents Definitions and Examples
Introduction
Dividing exponents can be a bit tricky, but once you understand the concept, it’s not so bad. In this blog post, we’ll go over the definition of dividing exponents and provide some examples to help you better understand the concept. So if you’re ready to learn more about this math topic, read on!
Multiplying and Dividing Exponents
When it comes to division, there are two different types of exponents you will encounter: positive and negative. Positive exponents, also known as base 10 exponents, simply imply the number of times a number is multiplied by itself. So, for example, 3 to the 4th power would be written 34 = 3 × 3 × 3 × 3. Negative exponents, on the other hand, tell you how many times to divide a number by itself. So, using our previous example, -4 would tell us to divide 3 by itself four times:
3-4 = _____
To solve this equation, we take our exponent (in this case -4) and move it to the denominator of a fraction with a 1 in the numerator. We then flip the fraction so that the exponent is now positive:
3-4 = 1/34
Now that we have a positive exponent, we can solve this equation like any other with positive exponents:
1/34 = 1/(3×3×3×3) = 1/27
Dividing Exponents
When dividing exponents with the same base, subtract the exponent of the divisor from the exponent of the dividend. This rule only applies when the base is the same for both dividend and divisor. When this rule is applied, it’s called division of exponents with like bases.
For example, 82 / 83 can be rewritten as 8-3 / 8-2. This can further be simplified to 8 / 8, which equals 1. So, 82 / 83 = 1.
Here’s another example. Suppose you wanted to divide 24-3 by 24-4. You could rewrite it as follows: 2-3 / 2-4 = 1/2 or simply 0.5
How to Divide Exponents?
To divide exponents, you need to have a basic understanding of what an exponent is. An exponent, or power, is a number that tells you how many times to multiply a base number by itself. The base number is the number that appears before the exponent.
For example, in the expression 8^2, 8 is the base number and 2 is the exponent. This expression can be read aloud as “8 to the second power” or “8 squared.” The answer to this expression would be 64 because 2 x 2 x 2 x 2 x 2 x 2 x 2 = 64. In this case, we would say that 8^2 = 64.
Now that we know what an exponent is, we can move on to division. When you divide two numbers with exponents that have the same base, you can simply subtract the exponents. For example:
8^4 / 8^2 = 8^4-2 = 8^2 = 64 / 8 = 8
In this equation, we are dividing 8 to the fourth power by 8 squared. Because they have the same base (8), we can subtract the exponents (4-2) to get our answer of 8 squared, or 64 / 8 which equals 8.
Dividing Exponents with Same Base
When you divide two exponential expressions that have the same base, you can use the following property:
For any positive real numbers a and b, and for any nonzero integer n:
a?/a? = a?-?
This property is also sometimes called “dividing exponents with the same base”, “division of exponents with same base”, or “index division”.
Dividing Exponents with Different Bases
When dividing exponents with different bases, you are really just performing division on the exponential values. In order to do this, you need to make sure that the divisor’s base is the same as the dividend’s base. This will make the division much easier.
To illustrate this concept, let’s look at an example:
Say we want to divide 85 by 23. In order to do this, we need to make sure that both numbers have the same base. The easiest way to do this is to convert both numbers into their equivalent exponential form. For 85, this would be 8^5 and for 23, this would be 2^3.
Now that we have both numbers in exponential form, we can simply divide the two values: 8^5 / 2^3 = (8/2)^5 = 4^5 = 1024.
So, when dividing exponents with different bases, all you need to do is convert both numbers into exponential form and then divide the two values.
Dividing Exponents with Coefficients
When dividing exponents with coefficients, the coefficients are divided first and then the exponents are subtracted. For example, if we divide 2^3 by 2^2, we would first divide 2 by 2 to get 1 and then subtract 3 from 2 to get -1. Therefore, the answer would be 1^-1 or 1/1.
Multiplying Exponential Terms
When it comes to dividing exponential terms, there are a few different rules that you need to remember. First, you can only divide exponents that have the same base. So, for example, you can divide 2^3 by 2^2, because both terms have a base of 2. However, you cannot divide 2^3 by 3^2, because the bases are different.
Second, when you divide exponents with the same base, you subtract the exponent of the term that you’re dividing by from the exponent of the term that you’re dividing into. So, using our previous example, we would have:
2^3 / 2^2 = 2^(3-2) = 2^1 = 2
Make sense? If not, don’t worry – we’ll go over some more examples below.
Finally, one last thing to remember is that when you have an exponent that is a negative number, this simply means that you need to take the reciprocal of the term. For example:
2^-3 = 1 / (2^3) = 1 / 8
How to Multiply and Divide Fractional Exponents?
To multiply exponents with the same base, simply add the exponents. So, remember when we learned how to multiply like terms? The same applies here. When the bases are the
same, we can add the exponents. For example:
(xy)^4 * (xz)^5 = x^9y^4z^5
Now let’s look at an example with different bases. This is where it gets a little bit more complicated, but we’ll go through it step by step.
(2x)^4 * (3y)^5 = 2x * 3y * 2x * 3y * 2x * 3y * 2x * 3y
= 6xy(2x)(2x)(2x)(2x)(3y)(3y)(3y)(3y)(3y)
= 6xy(8x^3)(81y^4)
= 6(8xy)(81xy) = 6(64xy) = 384xy
How to Multiply and Divide Exponents With Variables?
To multiply and divide exponents with variables, you need to use the laws of exponents. The law of exponents states that when you raise a power to a power, you multiply the exponents. So, if you have two variables, x and y, and you want to square them both, you would do x^2 * y^2 = (x*y)^2. This is because when you square a number, you’re really just multiplying it by itself. The same goes for cubing a number, which is just raising it to the third power.
When it comes to division, it’s a little bit different. You can’t just divide the exponents like you would with regular numbers. Instead, you need to use the law of exponents which states that when you divide two powers with the same base, you subtract the exponents. So, if you have x^5 / y^5, you would do (x/y)^(5-5), which would simplify to (x/y)^0 or just 1 since anything raised to the zero power is 1.
Keep in mind that these laws only work when both sides have the same base. So if you’re trying to multiply x^3 and y^4, you can’t just do (x*y)^7 because they don’t have the same base (in this case, 7 isn’t equal to 3+4).
Examples
There are two different types of division when it comes to exponents, and they each have their own set of rules. The first type is called long division, and it applies when the dividend has a greater value than the divisor. In this case, you divide the dividend by the divisor as you would normally divide any other number, and then you subtract the answer from the dividend. The remainder is your new dividend, and you continue this process until the remainder is less than the divisor. At that point, you can divide the final remainder by the divisor to get your answer.
The second type of division is called synthetic division, and it applies when the divisor has a greater value than the dividend. In this case, you write out the dividend as a series of terms with decreasing exponents until you reach the term with the lowest exponent. Then, you divide each term in the series by the divisor except for the term with the lowest exponent. The answer will be a series of terms with decreasing exponents that represents your quotient.
Conclusion
In mathematics, an exponent is a number that indicates how many times a particular value is to be used as a factor. Exponents are usually written as superscripts, for example 4^2 (read as “4 squared”), or x^3 (read as “x cubed”). In the expression 2^3, 2 is the base and 3 is the exponent. There are different rules for dividing exponents depending on whether the bases of the values being divided are the same or different. However, in all cases, when you divide two values with exponents, you will end up with a value that has an exponent equal to the difference between the exponents of the values being divided.
