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Properties of Exponents

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Properties of Exponents

Study Guide

Key Definition

The properties of exponents allow you to manipulate exponential expressions efficiently. For example, the product of powers property allows you to add exponents with the same base: $a^m \times a^n = a^{m+n}$

Important Notes

When multiplying like bases, add the exponents: $a^m \times a^n = a^{m+n}$
Raising a power to a power means multiplying the exponents: $(a^m)^n = a^{m \times n}$
Dividing like bases means subtracting the exponents: $\frac{a^m}{a^n} = a^{m-n}$
Any base to the zero power is one: $a^0 = 1$
Negative exponents represent reciprocal powers: $a^{-n} = \frac{1}{a^n}$

Mathematical Notation

$\times$ or $*$ represents multiplication

$\div$ or $/$ represents division

$a^n$ indicates a is raised to the power of n

$\frac{a}{b}$ represents a divided by b

Remember to use proper notation when solving problems

Why It Works

Understanding the properties of exponents helps simplify complex algebraic expressions by leveraging the rules of powers, such as $(a^m)^n = a^{m \times n}$.

Remember

Keep in mind the basic properties like $a^0 = 1$ and $a^{-n} = \frac{1}{a^n}$ as they form the foundation for more complex operations.

Quick Reference

Product of Powers:$a^m \times a^n = a^{m+n}$

Power of a Power:$(a^m)^n = a^{m \times n}$

Quotient of Powers:$\frac{a^m}{a^n} = a^{m-n}$

Zero Power:$a^0 = 1$

Negative Exponent:$a^{-n} = \frac{1}{a^n}$

Understanding Properties of Exponents

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Exponents can simplify repeated multiplication: $2^3 = 2 \times 2 \times 2$

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is $2^3 \times 2^4$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

If you double your allowance every week, starting with $2$ dollars (which is $2^1$), how much will you have after four weeks?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Can you find the result of $(3^2)^3$ using the power of a power property?

Challenge Quiz

Single Choice Quiz

Advanced

Simplify $\frac{5^7}{5^2}$.

Recap

Watch & Learn

Review key concepts and takeaways