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Powers of 10

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Powers of 10

Study Guide

Key Definition

In a base 10 number system, the expression $10^n$ represents 1 followed by $n$ zeros, while $10^{-n}$ represents a decimal with a 1 preceded by $n-1$ zeros.

Important Notes

Powers of 10 are frequently used in scientific notation.
Positive powers of 10 are whole numbers.
Negative powers of 10 are decimals less than 1.
Understanding powers of 10 helps simplify large and small numbers.
Being comfortable with powers of 10 is useful in many areas of math and science.

Mathematical Notation

$10^n$ represents 10 raised to the power of n

$10^{-n}$ represents a decimal form of power of 10

$10^a · 10^b = 10^{a+b}$

$(10^a)^b = 10^{a·b}$

Remember to use proper notation when solving problems

Why It Works

The powers of 10 simplify calculations by using exponential notation which makes it easier to handle large and small numbers with $10^n$ or $10^{-n}$.

Remember

Positive powers of 10 create large numbers, while negative powers create small decimals. Use $10^n$ and $10^{-n}$ to simplify expressions.

Quick Reference

10^0:$1$

10^1:$10$

10^{-1}:$0.1$

Understanding Powers of 10

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A power of 10 tells us how many times to multiply 10 by itself. For example, 10^3 means 10 × 10 × 10 = 1000. Notice that 10^0 = 1 because multiplying zero times leaves just 1.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is $10^3$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you are calculating the distance from Earth to a distant star which is $10^{16}$ kilometers.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

What numerical value does $10^{-3}$ represent, and why is it useful for measurements like millimeters?

Challenge Quiz

Single Choice Quiz

Advanced

Which of the following is equal to $10^5 \times 10^{-2}$?

Recap

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Review key concepts and takeaways