What Are Powers of Ten?

Our number system is a base-ten number system. This means that place values are built from powers of 10.

A power of ten is a number written using 10 as the base.

For example:

10¹ = 10

10² = 100

10³ = 1,000

10⁴ = 10,000

Powers of ten provide an efficient way to describe very large and very small numbers.

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Understanding Exponents

Consider:

10³

The number 10 is the base.

The number 3 is the exponent.

The exponent tells us how many times the base is used as a factor.

Therefore:

10³ = 10 × 10 × 10

10³ = 1,000

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Evaluating Powers of Ten

For positive whole-number exponents, evaluating powers of ten is straightforward.

10¹ = 10

10² = 10 × 10 = 100

10³ = 10 × 10 × 10 = 1,000

10⁴ = 10,000

10⁵ = 100,000

10⁶ = 1,000,000

Notice the pattern:

For positive whole-number powers of ten, the exponent tells us the number of zeros after 1.

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The Pattern of Zeros

Consider:

10¹ = 10

One zero.

10² = 100

Two zeros.

10³ = 1,000

Three zeros.

10⁶ = 1,000,000

Six zeros.

Therefore:

10⁸ = 100,000,000

because 1 is followed by eight zeros.

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Powers of Ten and Place Value

Each place in our number system is related to a power of ten.

Ones = 10⁰

Tens = 10¹

Hundreds = 10²

Thousands = 10³

Ten thousands = 10⁴

Hundred thousands = 10⁵

Millions = 10⁶

Billions = 10⁹

Trillions = 10¹²

Powers of ten are therefore built directly into our place-value system.

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Why Is 10⁰ Equal to 1?

Look at the pattern:

10³ = 1,000

10² = 100

10¹ = 10

Each time the exponent decreases by 1, divide by 10.

So:

10¹ ÷ 10 = 10⁰

and:

10 ÷ 10 = 1

Therefore:

10⁰ = 1

This is part of a general exponent rule:

For any nonzero number a:

a⁰ = 1

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Continuing Below Zero

Continue the same pattern:

10² = 100

10¹ = 10

10⁰ = 1

10⁻¹ = 0.1

10⁻² = 0.01

10⁻³ = 0.001

Every time the exponent decreases by 1, the value is divided by 10.

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Negative Powers of Ten

A negative exponent represents a reciprocal.

For example:

10⁻¹ = 1/10 = 0.1

10⁻² = 1/100 = 0.01

10⁻³ = 1/1000 = 0.001

10⁻⁶ = 1/1,000,000 = 0.000001

Negative powers of ten are especially useful for representing very small quantities.

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A Complete Pattern

Notice the pattern:

10⁴ = 10,000

10³ = 1,000

10² = 100

10¹ = 10

10⁰ = 1

10⁻¹ = 0.1

10⁻² = 0.01

10⁻³ = 0.001

Moving down one row:

÷ 10

Moving up one row:

× 10

This pattern connects exponents directly to place value.

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Moving Through Place Value

Consider the number:

4

Multiply by 10:

4 × 10 = 40

Multiply by 100:

4 × 100 = 400

Multiply by 1,000:

4 × 1,000 = 4,000

Using powers:

4 × 10¹ = 40

4 × 10² = 400

4 × 10³ = 4,000

Each multiplication moves the digit into a place with a value 10 times greater.

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A Better Way to Think About Decimal Movement

You may hear the shortcut:

"Move the decimal point."

This can be useful, but mathematically it is more accurate to think:

The digits change place value.

For example:

3.7 × 100 = 370

The decimal point has not physically moved. Instead, each digit now has a value 100 times greater.

Thinking in terms of place value helps avoid mistakes.

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Multiplying by 10

Consider:

24 × 10

Each digit becomes 10 times greater.

2 tens becomes 2 hundreds.

4 ones becomes 4 tens.

Therefore:

24 × 10 = 240

Another example:

6.35 × 10 = 63.5

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Multiplying by 100

Multiplying by:

100 = 10²

makes the number 100 times greater.

For example:

42 × 100 = 4,200

and:

3.75 × 100 = 375

Each digit shifts two place-value positions toward the larger place values.

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Multiplying by 1,000

Since:

1,000 = 10³

multiplication by 1,000 increases each digit's place value by three positions.

For example:

7.2 × 1,000 = 7,200

and:

0.043 × 1,000 = 43

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Multiplying by Powers of Ten

The same pattern works for any power of ten.

5.6 × 10¹ = 56

5.6 × 10² = 560

5.6 × 10³ = 5,600

5.6 × 10⁴ = 56,000

The exponent tells us how many place-value positions are involved.

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Dividing by 10

Division by 10 makes a number:

10 times smaller

For example:

350 ÷ 10 = 35

and:

8 ÷ 10 = 0.8

Each digit moves into a place worth one tenth as much.

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Dividing by 100

Since:

100 = 10²

division by 100 makes a number 100 times smaller.

For example:

4,500 ÷ 100 = 45

and:

72 ÷ 100 = 0.72

Each digit changes by two place-value positions.

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Dividing by 1,000

Since:

1,000 = 10³

division by 1,000 makes a number 1,000 times smaller.

For example:

82,000 ÷ 1,000 = 82

and:

5 ÷ 1,000 = 0.005

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Dividing by Powers of Ten

Consider:

6,400 ÷ 10¹ = 640

6,400 ÷ 10² = 64

6,400 ÷ 10³ = 6.4

6,400 ÷ 10⁴ = 0.64

Each increase in the divisor's exponent makes the result 10 times smaller.

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Multiplying by Negative Powers of Ten

Consider:

8 × 10⁻¹

Since:

10⁻¹ = 0.1

we have:

8 × 0.1 = 0.8

Similarly:

8 × 10⁻² = 0.08

8 × 10⁻³ = 0.008

A negative exponent makes the multiplier smaller than 1.

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Connecting Powers of Ten to Very Large Numbers

Powers of ten make large numbers easier to describe.

For example:

1 million = 10⁶

1 billion = 10⁹

1 trillion = 10¹²

So:

5 million = 5 × 10⁶

8 billion = 8 × 10⁹

3 trillion = 3 × 10¹²

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Connecting Powers of Ten to Very Small Numbers

Powers of ten are equally useful for tiny quantities.

For example:

one tenth = 10⁻¹

one thousandth = 10⁻³

one millionth = 10⁻⁶

one billionth = 10⁻⁹

These scales appear frequently in science and technology.

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Metric Prefixes and Powers of Ten

Many metric prefixes correspond directly to powers of ten.

For example:

kilo = 10³

centi = 10⁻²

milli = 10⁻³

micro = 10⁻⁶

nano = 10⁻⁹

This means:

1 kilometre = 10³ metres

1 millimetre = 10⁻³ metres

1 micrometre = 10⁻⁶ metres

1 nanometre = 10⁻⁹ metres

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Patterns in Powers of Ten

Look at:

10² = 100

10³ = 1,000

10⁴ = 10,000

Calculate:

10² × 10³

Using ordinary values:

100 × 1,000 = 100,000

But:

100,000 = 10⁵

Notice:

2 + 3 = 5

Therefore:

10² × 10³ = 10⁵

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Multiplying Powers with the Same Base

When multiplying powers with the same base, add the exponents.

10ᵃ × 10ᵇ = 10ᵃ⁺ᵇ

For example:

10⁴ × 10² = 10⁶

because:

4 + 2 = 6

Check:

10,000 × 100 = 1,000,000

and:

10⁶ = 1,000,000

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Another Multiplication Example

Simplify:

10³ × 10⁵

Add the exponents:

3 + 5 = 8

Therefore:

10³ × 10⁵ = 10⁸

Instead of writing and multiplying:

1,000 × 100,000

we can simply add the exponents.

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Dividing Powers with the Same Base

When dividing powers with the same base, subtract the exponents.

10ᵃ ÷ 10ᵇ = 10ᵃ⁻ᵇ

For example:

10⁷ ÷ 10³ = 10⁴

because:

7 − 3 = 4

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Why Subtraction Works

Consider:

10⁵ ÷ 10²

Expanded:

(10 × 10 × 10 × 10 × 10) ÷ (10 × 10)

Cancel two factors of 10.

Remaining:

10 × 10 × 10

Therefore:

10⁵ ÷ 10² = 10³

The exponent rule comes directly from repeated multiplication.

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Division Can Produce Negative Exponents

Consider:

10² ÷ 10⁵

Subtract exponents:

2 − 5 = −3

Therefore:

10² ÷ 10⁵ = 10⁻³

and:

10⁻³ = 0.001

This shows how negative exponents naturally arise from division.

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Powers Raised to Powers

Consider:

(10²)³

This means:

10² × 10² × 10²

Add the exponents:

2 + 2 + 2 = 6

Therefore:

(10²)³ = 10⁶

The general rule is:

(10ᵃ)ᵇ = 10ᵃᵇ

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Simplifying Calculations

Powers of ten can make calculations much easier.

Consider:

300 × 4,000

Rewrite:

300 = 3 × 10²

4,000 = 4 × 10³

Then:

(3 × 10²)(4 × 10³)

Multiply the ordinary numbers:

3 × 4 = 12

Combine the powers:

10² × 10³ = 10⁵

Therefore:

12 × 10⁵ = 1,200,000

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Another Simplification Example

Calculate:

6,000 × 20,000

Rewrite:

6,000 = 6 × 10³

20,000 = 2 × 10⁴

Then:

6 × 2 = 12

and:

10³ × 10⁴ = 10⁷

Therefore:

12 × 10⁷

which equals:

120,000,000

Using powers of ten reduces the amount of zero-counting required.

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Simplifying Division

Calculate:

800,000 ÷ 2,000

Rewrite:

800,000 = 8 × 10⁵

2,000 = 2 × 10³

Then:

8 ÷ 2 = 4

and:

10⁵ ÷ 10³ = 10²

Therefore:

4 × 10² = 400

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Worked Example 1: Evaluate a Power

Evaluate:

10⁷

A positive exponent of 7 means 1 followed by seven zeros.

Therefore:

10⁷ = 10,000,000

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Worked Example 2: Evaluate a Negative Power

Evaluate:

10⁻⁴

This means:

1/10⁴

Therefore:

10⁻⁴ = 1/10,000

or:

0.0001

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Worked Example 3: Multiplication

Calculate:

4.8 × 10³

Since:

10³ = 1,000

then:

4.8 × 1,000 = 4,800

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Worked Example 4: Division

Calculate:

75,000 ÷ 10³

Since:

10³ = 1,000

then:

75,000 ÷ 1,000 = 75

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Worked Example 5: Multiply Powers

Simplify:

10⁴ × 10⁶

Add exponents:

4 + 6 = 10

Therefore:

10⁴ × 10⁶ = 10¹⁰

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Worked Example 6: Divide Powers

Simplify:

10⁹ ÷ 10⁴

Subtract:

9 − 4 = 5

Therefore:

10⁹ ÷ 10⁴ = 10⁵

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Worked Example 7: Negative Exponents

Simplify:

10³ ÷ 10⁶

Subtract:

3 − 6 = −3

Therefore:

10³ ÷ 10⁶ = 10⁻³

or:

0.001

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Worked Example 8: Powers of Powers

Simplify:

(10³)⁴

Multiply the exponents:

3 × 4 = 12

Therefore:

(10³)⁴ = 10¹²

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Worked Example 9: Place Value

What happens to:

0.047

when multiplied by:

10³?

Since:

10³ = 1,000

calculate:

0.047 × 1,000 = 47

Each digit becomes 1,000 times greater in value.

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Worked Example 10: Simplifying a Calculation

Calculate:

5,000 × 300

Rewrite:

5,000 = 5 × 10³

300 = 3 × 10²

Then:

5 × 3 = 15

and:

10³ × 10² = 10⁵

So:

15 × 10⁵ = 1,500,000

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Powers of Ten and Scientific Notation

Powers of ten form the foundation of scientific notation.

For example, a large number such as:

450,000,000

can be expressed compactly using a coefficient and a power of ten.

A visualization helps show how the exponent controls the place-value shift:

The same idea works for very small numbers using negative exponents.

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Why Powers of Ten Are Useful in Science

Scientists regularly work with numbers that are extremely large or extremely small.

For example:

Astronomical distances may involve:

10⁹, 10¹², or much larger powers

Microscopic measurements may involve:

10⁻⁶ or 10⁻⁹

Powers of ten allow these quantities to be written, compared, and calculated efficiently.

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Powers of Ten in Computing

Computing involves enormous quantities of data and extremely short time intervals.

Powers of ten can help describe:

• storage capacities
• processing rates
• network speeds
• microscopic electronic components
• short processing times

For example:

10⁹

means one billion.

Meanwhile:

10⁻⁹

means one billionth.

The sign of the exponent makes an enormous difference.

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Comparing Powers of Ten

Compare:

10⁴ and 10⁷

Since:

10⁴ = 10,000

and:

10⁷ = 10,000,000

we know:

10⁷ > 10⁴

For positive powers of 10, the larger exponent produces the larger value.

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Comparing Negative Powers

Compare:

10⁻³ and 10⁻⁶

Convert:

10⁻³ = 0.001

10⁻⁶ = 0.000001

Therefore:

10⁻³ > 10⁻⁶

With negative powers, the exponent that looks "more negative" produces the smaller positive value.

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How Many Times Larger?

Compare:

10⁸

and:

10⁵

Divide:

10⁸ ÷ 10⁵ = 10³

Therefore:

10⁸ is 1,000 times larger than 10⁵

The difference between the exponents tells us the multiplicative difference.

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Another Scale Comparison

Compare:

10⁻²

and:

10⁻⁵

Divide:

10⁻² ÷ 10⁻⁵

Subtract exponents:

−2 − (−5) = 3

Therefore:

10³ = 1,000

So:

10⁻² is 1,000 times larger than 10⁻⁵

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Orders of Magnitude

Powers of ten are used to describe orders of magnitude.

Two quantities that differ by one power of ten differ by a factor of:

10

For example:

10⁵ and 10⁶

differ by one order of magnitude.

They differ by a factor of:

10

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Powers of Ten in Unit Conversions

Powers of ten make metric conversions easier.

For example:

1 km = 10³ m

Therefore:

4.2 km = 4.2 × 10³ m

= 4,200 m

Similarly:

1 mm = 10⁻³ m

Therefore:

7 mm = 7 × 10⁻³ m

= 0.007 m

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Real-World Example: Microscopy

Suppose a cell has a diameter of:

20 µm

Since:

1 µm = 10⁻⁶ m

then:

20 µm = 20 × 10⁻⁶ m

This can also be written:

2 × 10⁻⁵ m

Powers of ten make conversions between microscopic units much easier.

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Real-World Example: Astronomy

Suppose a distance is approximately:

150,000,000 km

This can be expressed as:

1.5 × 10⁸ km

Instead of repeatedly writing many zeros, powers of ten provide a compact representation.

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Estimating with Powers of Ten

Suppose you want to estimate:

9,800 × 21,000

Round:

9,800 ≈ 10,000 = 10⁴

21,000 ≈ 20,000 = 2 × 10⁴

Then:

10⁴ × 2 × 10⁴

= 2 × 10⁸

So the product is approximately:

200,000,000

This allows a quick reasonableness check.

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Using Powers of Ten to Check Answers

Suppose a calculation involving:

3,000 × 40,000

produces an answer of:

120,000

Is that reasonable?

Estimate the powers:

3,000 ≈ 10³

40,000 ≈ 10⁴

Their product should be roughly:

10⁷

So an answer around:

100,000 = 10⁵

is far too small.

The correct calculation is:

3,000 × 40,000 = 120,000,000

Powers of ten help identify errors quickly.

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Common Mistakes

Mistake 1: Multiplying the base by the exponent

Incorrect:

10³ = 10 × 3 = 30

Correct:

10³ = 10 × 10 × 10 = 1,000

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Mistake 2: Thinking 10⁰ = 0

Correct:

10⁰ = 1

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Mistake 3: Thinking a negative exponent makes the number negative

Incorrect:

10⁻³ = −1,000

Correct:

10⁻³ = 0.001

The negative sign belongs to the exponent, not the value.

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Mistake 4: Adding exponents during addition

The rule:

10² × 10³ = 10⁵

works for multiplication.

But:

10² + 10³ ≠ 10⁵

Instead:

100 + 1,000 = 1,100

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Mistake 5: Multiplying exponents when multiplying powers

Incorrect:

10² × 10³ = 10⁶

Correct:

10² × 10³ = 10⁵

Add exponents when multiplying powers with the same base.

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Mistake 6: Losing place value when multiplying decimals

Always check whether the result should become larger or smaller.

Multiplication by:

10³

should make a positive number 1,000 times larger.

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Error Analysis

A student writes:

10⁵ ÷ 10² = 10²⋅⁵

This is incorrect.

When dividing powers with the same base, subtract the exponents:

5 − 2 = 3

Therefore:

10⁵ ÷ 10² = 10³

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Another Error Analysis

A student says:

10⁻⁴ is greater than 10⁻² because 4 is greater than 2.

Write the values:

10⁻⁴ = 0.0001

10⁻² = 0.01

Therefore:

10⁻⁴ < 10⁻²

With negative exponents, a more negative exponent represents a smaller positive value.

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A Reliable Strategy for Evaluating Powers of Ten

For a positive exponent:

Step 1: Write 1.

Step 2: Add the number of zeros indicated by the exponent.

Example:

10⁵ = 100,000

For a zero exponent:

10⁰ = 1

For a negative exponent:

Step 1: Think of the reciprocal.

Step 2: Convert to a decimal if necessary.

Example:

10⁻⁴ = 1/10⁴ = 0.0001

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A Reliable Strategy for Multiplication and Division

When multiplying by a positive power of ten:

The number becomes larger.

When dividing by a positive power of ten:

The number becomes smaller.

When multiplying powers with the same base:

Add exponents.

When dividing powers with the same base:

Subtract exponents.

When raising a power to another power:

Multiply exponents.

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Did You Know?

Powers of ten allow us to describe an enormous range of scales using the same mathematical system.

For example:

10⁻⁹ m is the scale of nanometres.

10⁰ m is one metre.

10³ m is one kilometre.

10⁶ m is one thousand kilometres.

Much larger powers can describe planetary and astronomical distances.

The same base-ten pattern connects microscopic measurements with enormous astronomical quantities.

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Key Terms

• Base: Number being raised to a power.
• Exponent: Number showing how many times the base is used as a factor.
• Power: Expression consisting of a base and exponent.
• Power of ten: Expression such as 10³ or 10⁻⁶.
• Place value: Value of a digit determined by its position.
• Positive exponent: Exponent greater than zero.
• Zero exponent: Exponent of zero; for nonzero bases, the value is 1.
• Negative exponent: Exponent representing a reciprocal.
• Reciprocal: Multiplicative inverse of a number.
• Scientific notation: Method for expressing numbers using a coefficient and a power of ten.
• Order of magnitude: Scale of a quantity described by a power of ten.
• Metric prefix: Prefix such as kilo, milli, micro, or nano representing a power of ten.

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Key Rules

Positive powers:

10ⁿ = 1 followed by n zeros, for positive whole-number n.

Zero power:

10⁰ = 1

Negative powers:

10⁻ⁿ = 1/10ⁿ

Multiplying powers:

10ᵃ × 10ᵇ = 10ᵃ⁺ᵇ

Dividing powers:

10ᵃ ÷ 10ᵇ = 10ᵃ⁻ᵇ

Power of a power:

(10ᵃ)ᵇ = 10ᵃᵇ

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Key Takeaways

• Powers of ten are fundamental to our base-ten number system.
• The base is 10 and the exponent describes the power.
• A positive whole-number exponent tells how many factors of 10 are multiplied together.
• Positive powers of ten produce 1 followed by a predictable number of zeros.
• Each increase of 1 in the exponent multiplies the value by 10.
• Each decrease of 1 in the exponent divides the value by 10.
• 10⁰ equals 1.
• Negative powers of ten represent values between 0 and 1.
• 10⁻¹ equals 0.1.
• 10⁻² equals 0.01.
• 10⁻³ equals 0.001.
• Powers of ten correspond directly to place value.
• Multiplying by 10, 100, or 1,000 increases place values by one, two, or three positions.
• Dividing by powers of ten decreases place values in the same systematic way.
• Thinking about digits changing place value is more reliable than simply memorizing "move the decimal."
• When multiplying powers with the same base, add the exponents.
• When dividing powers with the same base, subtract the exponents.
• When raising a power to another power, multiply the exponents.
• Exponent rules for multiplication should not be incorrectly applied to addition.
• Powers of ten can simplify calculations containing large numbers and many zeros.
• They can also be used to estimate answers and check whether calculations are reasonable.
• Metric prefixes such as kilo, milli, micro, and nano correspond to specific powers of ten.
• Powers of ten are essential for scientific notation.
• Orders of magnitude allow quantities at very different scales to be compared efficiently.
• Powers of ten are widely used in mathematics, science, engineering, computing, medicine, and technology.
• Understanding powers of ten provides the foundation for working confidently with scientific notation, metric conversions, very large numbers, very small numbers, and exponent rules.