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# Factoring Exponentials

November 21, 2023 written by Rida Mirza

An exponential expression typically looks like this: a^b, where ‘a’ is the base, and ‘b’ is the exponent. It represents the repeated multiplication of ‘a’ by itself ‘b’ times.

In this article, we will discuss the process of factoring exponentials, and provide solved examples.

## Factoring Exponents

Factoring exponentials involves expressing them as a product of simpler exponential expressions. To do this, we need to find common factors in the terms and simplify.

### Example

Factor the expression 2^3 * 2^4.

Here, we have two terms with the same base (2). We can add the exponents to simplify: 2^(3+4) = 2^7.

So, 2^3 * 2^4 = 2^7.

## Factoring Exponentials with Different Bases

Sometimes, you might encounter exponentials with different bases. In such cases, you can factor them by recognizing common factors and simplifying them.

### Example

Factor the expression 3^2 * 2^2.

Here, we can’t combine the bases, but we can simplify each term individually:

3^2 * 2^2 = 9 * 4 = 36.

So, 3^2 * 2^2 = 36.

## Factoring Exponentials with Negative Exponents

Negative exponents might seem tricky, but they follow a straightforward rule:

a^(-b) = 1 / (a^b)

We use this rule to factor exponentials with negative exponents.

### Example

Factor the expression 4^(-2).

Apply the rule:

4^(-2) = 1 / 4^2 = 1 / 16.

So, 4^(-2) = 1/16.

## Factoring Exponentials with Variables

Factoring exponentials with variables is a bit more complex, but it follows similar principles. You factor by finding common bases and simplifying the exponents.

### Example

Factor the expression x^2 * x^3.

Here, both terms have the same base (x), so we can add the exponents:

x^(2+3) = x^5.

So, x^2 * x^3 = x^5.

## FAQs

### What is the rule for factoring exponentials with the same base?

To factor exponentials with the same base, add their exponents.

### How do you factor exponentials with different bases?

You can’t combine them, but you can simplify each term individually.

### What’s the rule for factoring exponentials with negative exponents?

A^(-b) = 1 / (a^b).

### Can you factor exponentials with variables?

Yes, you can factor exponentials with variables by simplifying the exponents and identifying common bases.

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