Hi. This page will be about exponent laws and algebra. This guide is more suited for a high school audience and can be used as a refresher for pre-calculus and calculus students.

**Table Of Contents**

**What Is An Exponent?**

Suppose I want to multiply the number 2 by itself 4 times. I can write this as . If I wanted to multiply 2 by itself 20 times, that would take up a lot of space. Instead of doing with 20 twos and 19 multiplication signs we can expressed this as . The 2 is a base while the superscripted number 20 is the exponent.

If I wanted to multiply 5 by itself 4 times I can write or simply . The 4 here is like a counter of how many times the base number 5 is multiplied by itself. An alternate view would be . In this case I can add the exponents 2 and 2 to get 4 as long as the bases are the same.

**Exponent Laws**

Mathematics contains a lot of rules (axioms). Some could say that mathematics is like a language. Here are the rules/laws of exponents. ( and are typically whole numbers)

Multiplying Numbers of The Same Base:

Dividing Numbers of The Same Base:

Power of A Power (Power Rule):

Zero Exponent: because

Negative Exponents: and

Negative Exponents (Version 2): and

**Notes**

There are times when you may have to apply multiple exponent laws. For example we could have . This example applies the negative exponent then the power rule.

It is important to note that that which is different from . The exponent 2 is applied to inside the bracket in while the exponent 2 is applied to only in .

When dealing with square roots, remember that . In general, the n root .

Recall that . A special case of this would be .

**Examples**

Here are some examples using various exponent laws.

Example One

Convert from negative exponents to positive exponents.

Answer:

Example Two

Simplify .

Answer:

Example Three

Evaluate .

Answer:

**Practice Problems**

Here are some practice problems to test your understanding and build your skills. The answers are in the next section.

1) Convert from negative exponents to positive exponents.

2) Evaluate .

3) Convert from negative exponents to positive exponents.

4) Convert the fractions and into non-fractions with negative exponents.

5) Simplify .

6) Simplify .

7) Evaluate .

**Answers**

1)

2)

3)

4) and

5)

6)

7)