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Negative Exponents

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Negative Exponents

Study Guide

Key Definition

A negative exponent indicates how many times to divide the base number. For example, $4^{-2}$ means $\frac{1}{4^2}$.

Important Notes

A negative exponent means the reciprocal of the base raised to the opposite positive exponent: $a^{-n} = \frac{1}{a^n}$
Negative exponents can be converted to positive exponents by taking the reciprocal.
The expression $(\frac{1}{a})^n$ is equivalent to $a^{-n}$.
Zero exponent: $a^0 = 1$ for any non-zero $a$.
Be sure to apply the negative exponent rule across entire expressions.

Mathematical Notation

$^$ denotes an exponent

$\frac{1}{a^n}$ is the reciprocal of $a^n$

$\times$ represents multiplication

$\div$ represents division

Remember to use proper notation when solving problems

Why It Works

Negative exponents work by transforming multiplications into divisions, effectively using the reciprocal of the base: $a^{-n} = \frac{1}{a^n}$.

Remember

To convert negative exponents to positive, take the reciprocal: $a^{-n} = \frac{1}{a^n}$.

Quick Reference

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

Zero Exponent:$a^0 = 1$

Understanding Negative Exponents

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Negative exponents tell us how many times to divide the base: e.g., $4^{-2} = \frac{1}{4^2} = \frac{1}{16}$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is $2^{-3}$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you have a game that doubles in score each level, but level 3 shows a score of $2^{-3}$. What fraction of the maximum score of that level does this represent?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Consider the expression $5^{-2} \times 3^{-1}$. How can you simplify it?

Challenge Quiz

Single Choice Quiz

Advanced

Simplify $(\frac{2}{3})^{-3}$

Recap

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Review key concepts and takeaways