# Algebra – Working With Exponents

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.

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.

Example Two

Simplify .

Example Three

Evaluate .

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 .

1)

2)

3)

4) and

5)

6)

7)