Exponents And Small Number

Unit: Exponents & Powers

Chapter: Exponents & Small Numbers

Reference: – What are Small Numbers, Negative Exponents for Small Numbers, Standard Form for Small Numbers (Scientific Notation), Converting Decimal to Standard Form, Converting Standard Form to Decimal, Comparing Very Small Numbers, Ordering Small Numbers, Real-Life Examples (Microscopic Sizes), Solved Examples, Odd-One-Out Problems, Common Mistakes

After studying this chapter, you should be able to understand:

• How to Write Very Small Numbers Using Negative Exponents
• How to Convert Small Decimals to Standard Form (A × 10⁻ⁿ)
• How to Convert Standard Form with Negative Exponents to Decimals
• How to Compare and Order Very Small Numbers

Introduction to Exponents & Small Numbers

Definition

Small numbers are numbers between 0 and 1 (like 0.001, 0.00005, 0.0000002). Using negative exponents, we can write these small numbers in a compact form called standard form (scientific notation). For example, 0.001 = 1 × 10⁻³, where the negative exponent tells us how many places the decimal point moved to the right.

When we write small numbers using exponents, we essentially ask:

"How can I express this tiny number in a shorter, easier-to-read way?"

Negative exponents are the key to representing very small numbers efficiently.

Importance of Writing Small Numbers with Exponents

• Used in science (size of bacteria, viruses, atoms)
• Used in medicine (doses of medicine, cell sizes)
• Used in physics (wavelengths of light, atomic particles)
• Used in chemistry (molecular sizes, concentrations)
• Makes calculations with very small numbers easier

Example

The size of a red blood cell is about 0.000007 m. In standard form: 7 × 10⁻⁶ m.
The mass of a dust particle is about 0.0000000007 kg. In standard form: 7 × 10⁻¹⁰ kg.

Subtopics

1. Understanding Negative Exponents for Small Numbers

A negative exponent tells us that the number is less than 1.

Pattern:

Key Observation: The negative exponent tells the number of decimal places after the decimal point before the first non-zero digit.

Example: 10⁻³ = 0.001 (3 decimal places before the 1)

2. Standard Form for Small Numbers (Scientific Notation)

A very small number is written in standard form as: A × 10^(-n) where:

• 1 ≤ A < 10 (A is a number between 1 and 10, can be a decimal)
• n is a positive integer (the number of places the decimal moved)

Rules for Writing Small Numbers in Standard Form:

Step 1: Move the decimal point to the right until you have a number between 1 and 10.

Step 2: Count how many places you moved the decimal point. That number becomes n.

Step 3: Write the number as A × 10^(-n).

Example 1: Write 0.005 in standard form

0.005 → move decimal 3 places right → 5 → between 1 and 10
0.005 = 5 × 10⁻³

Example 2: Write 0.00042 in standard form

0.00042 → move decimal 4 places right → 4.2 → between 1 and 10
0.00042 = 4.2 × 10⁻⁴

Example 3: Write 0.0000003 in standard form

0.0000003 → move decimal 7 places right → 3 → between 1 and 10
0.0000003 = 3 × 10⁻⁷

Example 4: Write 0.0000105 in standard form

0.0000105 → move decimal 5 places right → 1.05 → between 1 and 10
0.0000105 = 1.05 × 10⁻⁵

3. Converting Standard Form (Small Numbers) to Decimal

To convert A × 10^(-n) to decimal form, move the decimal point n places to the left (add zeros as needed).

Example 1: 3 × 10⁻⁴ = 0.0003 (move decimal 4 places left from 3.0)

Example 2: 2.5 × 10⁻³ = 0.0025

Example 3: 1.23 × 10⁻⁵ = 0.0000123

Example 4: 9 × 10⁻⁶ = 0.000009

4. Comparing Very Small Numbers

When comparing numbers in standard form with negative exponents, remember: More negative exponent means smaller number.

Rule: Compare the exponents first. The number with the larger (less negative) exponent is larger.

Order from largest to smallest: 10⁻² > 10⁻³ > 10⁻⁴

Examples:

Comparing with different A values (same exponent): Compare A values.

Example: Which is larger: 3 × 10⁻⁴ or 5 × 10⁻⁴?
Exponents are same (-4). Compare 3 and 5 → 5 × 10⁻⁴ is larger.

Comparing with different exponents: Larger exponent (less negative) = larger number.

Example: Which is larger: 2 × 10⁻³ or 8 × 10⁻⁵?

2 × 10⁻³ = 0.002, 8 × 10⁻⁵ = 0.00008 → 2 × 10⁻³ is larger because exponent -3 > -5

5. Ordering Small Numbers from Smallest to Largest

To order small numbers, write them all in standard form with the same exponent, then compare.

Example: Order 0.0005, 0.00003, 0.0002 from smallest to largest.

Write all in standard form: 5 × 10⁻⁴, 3 × 10⁻⁵, 2 × 10⁻⁴

Convert to same exponent (say 10⁻⁵): 50 × 10⁻⁵, 3 × 10⁻⁵, 20 × 10⁻⁵

Now order A values: 3, 20, 50 → 3 × 10⁻⁵, 20 × 10⁻⁵, 50 × 10⁻⁵

So smallest to largest: 0.00003, 0.0002, 0.0005

Solved Examples

Example 1 – Write in Standard Form: Write 0.00007 in standard form.

Solution: Move decimal 5 places right → 7 → 7 × 10⁻⁵

Answer: 7 × 10⁻⁵

Example 2 – Write in Standard Form: Write 0.0000234 in standard form.

Solution: Move decimal 5 places right → 2.34 → 2.34 × 10⁻⁵

Answer: 2.34 × 10⁻⁵

Example 3 – Convert to Decimal: Write 4.5 × 10⁻⁶ as a decimal.

Solution: Move decimal 6 places left: 0.0000045

Answer: 0.0000045

Example 4 – Convert to Decimal: Write 1.23 × 10⁻⁴ as a decimal.

Solution: Move decimal 4 places left: 0.000123

Answer: 0.000123

Example 5 – Compare: Which is larger: 6 × 10⁻⁵ or 2 × 10⁻⁴?

Solution: Exponents: -5 and -4. Since -4 > -5, 2 × 10⁻⁴ is larger.

Answer: 2 × 10⁻⁴

Example 6 – Order: Order from smallest to largest: 3 × 10⁻⁴, 8 × 10⁻⁶, 5 × 10⁻⁵

Solution: Write all with exponent 10⁻⁶: 300 × 10⁻⁶, 8 × 10⁻⁶, 50 × 10⁻⁶
Order A values: 8, 50, 300
Smallest to largest: 8 × 10⁻⁶, 5 × 10⁻⁵, 3 × 10⁻⁴

Answer: 8 × 10⁻⁶, 5 × 10⁻⁵, 3 × 10⁻⁴

Common Mistakes to Avoid

Mistake 1 – Moving decimal, the wrong way
0.0005 = 5 × 10⁻⁴ (move RIGHT 4 places), NOT 5 × 10⁴.
Correct understanding: Small numbers (between 0 and 1) use negative exponents.

Mistake 2 – Counting decimal places incorrectly
0.0003 has 4 decimal places before 3? Actually 0.0003 = 3 × 10⁻⁴ (3 is in the 4th decimal place).
Correct understanding: Count how many places you move the decimal to get between 1 and 10.

Mistake 3 – Forgetting that A must be between 1 and 10
Writing 0.5 × 10⁻³ is incorrect standard form.
Correct understanding: 0.5 × 10⁻³ = 5 × 10⁻⁴ (adjust exponent).

Mistake 4 – Confusing 10⁻⁴ with 10⁴
10⁻⁴ = 0.0001, 10⁴ = 10,000 (very different!).
Correct understanding: Negative exponent = number less than 1.

Mistake 5 – Comparing negative exponents incorrectly
Thinking 10⁻⁵ > 10⁻⁴ because 5 > 4. Wrong! -5 is less than -4.
Correct understanding: 10⁻⁴ = 0.0001, 10⁻⁵ = 0.00001, so 10⁻⁴ > 10⁻⁵.

Mistake 6 – Adding zeros when converting to decimal
3 × 10⁻⁵ = 0.00003 (have 4 zeros before the 3? No, 5 decimal places total: 0.00003).
Correct understanding: Move decimal 5 places left from 3.0 → 0.00003.

Quick Reference Summary

Small Numbers (0 to 1): Written as A × 10^(-n) where 1 ≤ A < 10, n positive integer

Converting Decimal to Standard Form: Move decimal RIGHT until A is between 1 and 10 → n = number of moves → A × 10^(-n)

Converting Standard Form to Decimal: Move decimal LEFT n places

Comparing: Larger (less negative) exponent = larger number. If exponents same, compare A.

Common Small Numbers:
0.1 = 1 × 10⁻¹
0.01 = 1 × 10⁻²
0.001 = 1 × 10⁻³
0.0001 = 1 × 10⁻⁴
0.00001 = 1 × 10⁻⁵

Real-Life Examples:
Red blood cell: 7 × 10⁻⁶ m
Virus: 1 × 10⁻⁷ m
Atom: 1 × 10⁻¹⁰ m