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# Multiplying Exponents

When we multiply exponents, we multiply a number’s power.

Multiplying exponents is an essential mathematical concept that can seem complex initially, but with a solid understanding of the rules and principles, it becomes much more straightforward.

This article will cover the basic rules of multiplying exponents and how to apply them, including examples. We will also explore more advanced concepts, such as multiplying exponents with different bases and how exponents relate to roots. By the end of this article, you will have a solid understanding of how to multiply exponents and be able to apply these concepts to solve problems in various fields.

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For example, if we have the expression (2^3) x (2^4), we are multiplying 2 raised to the power of 3, with 2 raised to the power of 4. To solve this expression, we would use the rule for multiplying exponents, which states that when we have the same base, we add the exponents. Therefore, our answer would be 2^7, or 128.

It’s also important to note that the rule for multiplying exponents is different when we have different bases. In that case, we would just multiply the bases and keep the exponents as they are.

## Rules of Multiplying Exponents

The rules for multiplying exponents are as follows:

### 1. Product of Power Rules

This rule states that when multiplying exponents with the same base, we add the exponents.

For example, (2^3) x (2^4) = 2^(3+4) = 2^7 = 128. This means we are multiplying 2 raised to the power of 3, with 2 raised to the power of 4.

### 2. Power of a Power Rule

This rule states that when raising an exponent to another exponent, we multiply the exponents.

For example, (2^3)^2 = 2^(3*2) = 2^6 = 64. This means that 2 raised to the power of 3, raised to the power of 2, is equal to 2 raised to the power of 6.

### 3. Power of a Product Rule

This rule states that when raising a product of numbers to an exponent, we raise each factor to that exponent.

For example, (3×5)^2 = 3^2 x 5^2 = 9×25 = 225.

### 4. Quotient of Powers Rule

This rule states that when dividing exponents with the same base, we subtract the exponents.

For example, (8^3) / (8^2) = 8^(3-2) = 8^1 = 8. This means we are dividing 8 raised to the power of 3, by 8 raised to the power of 2.

### 5. Power of Quotient Rule

This rule states that when raising a quotient of numbers to an exponent, we raise each factor to that exponent.

For example, (16/4)^3 = (4^2) / (2^2) = 4^3 / 2^3 = 64/8 = 8.

To solve this question just change the different bases and perform the operation like 16 will become 2 raised to power 4 and 4 will become 2 raised to power 2. Then solve with the exponent law of division.

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It’s essential to keep in mind that these rules are only applicable when the bases are the same. When the bases are different, the rules are different and as follows:

#### 1. Multiplication of Exponents

To multiply exponents with different bases, we simply multiply them together and keep them as they are.

#### 2. Division of Exponents

To divide exponents with different bases, we divide the bases and subtract the exponents.

It’s important to remember that these rules only apply when the bases are different; if the bases are the same, you must use the previously mentioned rules.

Also, when working with the division of exponents, keep in mind that if the subtraction of exponents results in a negative exponent, it is equivalent to 1 over the corresponding expression with a positive exponent.

Gloria Mathew writes on math topics for K-12. A trained writer and communicator, she makes math accessible and understandable to students at all levels. Her ability to explain complex math concepts with easy to understand examples helps students master math. LinkedIn

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