Powers of Ten

Fifth Grade Mathematics - Complete Guide

🔢 Understanding Powers of Ten

What are Powers of Ten?

Powers of ten are numbers where 10 is multiplied by itself a certain number of times[web:65][web:66]. They are written using exponential notation[web:68].

Exponential Notation[web:65][web:66]:

\( 10^n \) where \( n \) is the exponent

Read as: "10 to the power of \( n \)" or "10 to the \( n \)th power"

Parts of Exponential Form:

\( 10^5 \)

↓

Base

(The number being multiplied)

↓

Exponent

(How many times to multiply)

Three Forms of Numbers[web:65]:

Exponential Form	Expanded Form	Standard Form	Word Form
\( 10^0 \)	1	1	One
\( 10^1 \)	10	10	Ten
\( 10^2 \)	\( 10 \times 10 \)	100	One Hundred
\( 10^3 \)	\( 10 \times 10 \times 10 \)	1,000	One Thousand
\( 10^4 \)	\( 10 \times 10 \times 10 \times 10 \)	10,000	Ten Thousand
\( 10^5 \)	\( 10 \times 10 \times 10 \times 10 \times 10 \)	100,000	One Hundred Thousand
\( 10^6 \)	\( 10 \times 10 \times 10 \times 10 \times 10 \times 10 \)	1,000,000	One Million

Key Pattern Discovery[web:66][web:67]:

The exponent tells you how many ZEROS come after the 1

Formula: \( 10^n = \) 1 followed by \( n \) zeros[web:66]

🧮 Evaluate Powers of Ten

What Does "Evaluate" Mean?

To evaluate a power of ten means to find its value in standard form (the actual number)[web:65][web:67].

Method 1: Counting Zeros[web:66][web:67]:

Quick Method:

Write 1, then add as many zeros as the exponent

Examples:

\( 10^2 = 100 \) (1 with 2 zeros)
\( 10^3 = 1{,}000 \) (1 with 3 zeros)
\( 10^4 = 10{,}000 \) (1 with 4 zeros)
\( 10^5 = 100{,}000 \) (1 with 5 zeros)
\( 10^6 = 1{,}000{,}000 \) (1 with 6 zeros)

Method 2: Repeated Multiplication[web:65]:

Step-by-Step Method:

Multiply 10 by itself the number of times shown by the exponent

Example: Evaluate \( 10^4 \)

Step 1: The exponent is 4, so multiply 10 four times

Step 2: \( 10 \times 10 \times 10 \times 10 \)

Step 3: \( 10 \times 10 = 100 \)

Step 4: \( 100 \times 10 = 1{,}000 \)

Step 5: \( 1{,}000 \times 10 = 10{,}000 \)

Answer: \( 10^4 = 10{,}000 \)

Special Cases[web:66]:

Case 1: Zero Exponent

\( 10^0 = 1 \)

Rule: Any number (except 0) raised to the power of 0 equals 1[web:66]

Case 2: Exponent of One

\( 10^1 = 10 \)

Rule: Any number raised to the power of 1 equals itself[web:66]

Practice Problems:

\( 10^7 = \) ?

Answer: 10,000,000

\( 10^8 = \) ?

Answer: 100,000,000

\( 10^9 = \) ?

Answer: 1,000,000,000

\( 10^{10} = \) ?

Answer: 10,000,000,000

✍️ Write Powers of Ten with Exponents

Converting to Exponential Form[web:65][web:66]:

When you see a standard number (like 1,000), you can write it as a power of ten by counting the zeros.

Method 1: Count the Zeros[web:67]:

Steps:

Look at the number in standard form
Count how many zeros come after the 1
Write 10 with the number of zeros as the exponent

Examples:

100 has 2 zeros → \( 10^2 \)

1,000 has 3 zeros → \( 10^3 \)

10,000 has 4 zeros → \( 10^4 \)

100,000 has 5 zeros → \( 10^5 \)

1,000,000 has 6 zeros → \( 10^6 \)

Method 2: Writing Large Numbers as Powers of Ten[web:65][web:66]:

For numbers with other digits (not just 1 and zeros):

Number = Coefficient × \( 10^n \)

Example 1: Write 45,000 using a power of ten[web:65]

Step 1: Identify the non-zero digits: 45

Step 2: Count the trailing zeros: 3 zeros

Step 3: Write as: \( 45 \times 10^3 \)

Example 2: Write 7,000,000 using a power of ten

Step 1: Identify the non-zero digits: 7

Step 2: Count the trailing zeros: 6 zeros

Step 3: Write as: \( 7 \times 10^6 \)

Quick Conversion Table:

Standard Form	Number of Zeros	Exponential Form
10	1	\( 10^1 \)
100	2	\( 10^2 \)
1,000	3	\( 10^3 \)
10,000	4	\( 10^4 \)
100,000	5	\( 10^5 \)
1,000,000	6	\( 10^6 \)
10,000,000	7	\( 10^7 \)
100,000,000	8	\( 10^8 \)
1,000,000,000	9	\( 10^9 \)

🔄 Patterns with Powers of Ten

Multiplying by Powers of Ten[web:65]:

Pattern Discovery:

When multiplying by \( 10^n \), move the decimal point \( n \) places to the RIGHT

Examples:

\( 67 \times 10^1 = 670 \) (move 1 place right)
\( 67 \times 10^2 = 6{,}700 \) (move 2 places right)
\( 67 \times 10^3 = 67{,}000 \) (move 3 places right)

Dividing by Powers of Ten[web:65]:

Pattern Discovery:

When dividing by \( 10^n \), move the decimal point \( n \) places to the LEFT

Examples:

\( 702{,}000 \div 10^1 = 70{,}200 \) (move 1 place left)
\( 702{,}000 \div 10^2 = 7{,}020 \) (move 2 places left)
\( 702{,}000 \div 10^3 = 702 \) (move 3 places left)[web:65]

Place Value Connection[web:65]:

Each place value position is 10 times greater than the position to its right:

Ones place = \( 10^0 = 1 \)
Tens place = \( 10^1 = 10 \)
Hundreds place = \( 10^2 = 100 \)
Thousands place = \( 10^3 = 1{,}000 \)
Ten Thousands place = \( 10^4 = 10{,}000 \)
Hundred Thousands place = \( 10^5 = 100{,}000 \)
Millions place = \( 10^6 = 1{,}000{,}000 \)

📐 Essential Formulas for Powers of Ten

Formula 1: Basic Power of Ten

\( 10^n = \underbrace{10 \times 10 \times 10 \times \ldots \times 10}_{n \text{ times}} \)

Formula 2: Counting Zeros Rule

\( 10^n = 1 \) followed by \( n \) zeros

Formula 3: Standard to Exponential

If standard form has \( n \) zeros, then exponential form is \( 10^n \)

Formula 4: Multiplication Pattern

\( a \times 10^n = a \) with \( n \) zeros added

Formula 5: Special Exponents

\( 10^0 = 1 \)

\( 10^1 = 10 \)

Formula 6: Product of Powers of Ten

\( 10^a \times 10^b = 10^{a+b} \)

Example: \( 10^2 \times 10^3 = 10^5 = 100{,}000 \)

📖 Word Problems with Powers of Ten

Real-World Applications[web:68]:

Example 1: Distance Problem

Problem: The distance from Earth to the Moon is approximately \( 3.84 \times 10^5 \) kilometers. What is this distance in standard form?

Solution:

Step 1: Evaluate \( 10^5 = 100{,}000 \)

Step 2: Multiply: \( 3.84 \times 100{,}000 = 384{,}000 \)

Answer: 384,000 kilometers

Example 2: Population Problem

Problem: A city has a population of 5,000,000. Write this using a power of ten.

Solution:

Step 1: Identify non-zero digits: 5

Step 2: Count trailing zeros: 6 zeros

Step 3: Write as: \( 5 \times 10^6 \)

Answer: \( 5 \times 10^6 \)

Example 3: Comparison Problem

Problem: Which is greater: \( 10^4 \) or \( 10^3 \)?

Solution:

\( 10^4 = 10{,}000 \)

\( 10^3 = 1{,}000 \)

Since \( 10{,}000 > 1{,}000 \)

Answer: \( 10^4 \) is greater

📋 Quick Reference Chart

Exponential Form	Standard Form	Word Form	Number of Zeros
\( 10^0 \)	1	One	0
\( 10^1 \)	10	Ten	1
\( 10^2 \)	100	Hundred	2
\( 10^3 \)	1,000	Thousand	3
\( 10^4 \)	10,000	Ten Thousand	4
\( 10^5 \)	100,000	Hundred Thousand	5
\( 10^6 \)	1,000,000	Million	6
\( 10^9 \)	1,000,000,000	Billion	9