Use of Exponents to Express Large and Small Numbers in Standard Form

For large Numbers

1. Introduction

• Numbers that are very large or very small are written in standard form (scientific notation).
• Standard form makes calculations and comparisons easier.
• A number in standard form is written as: a×10ⁿ  where 1≤a<10 and $n$ is an integer.

2. When the Exponent is Positive

A positive exponent means the number is greater than 10.

• The exponent tells how many places the decimal moves to the right.

Steps to Convert Large Numbers to Standard Form

1. Identify the first non-zero digit.
2. Place a decimal after the first digit.
3. Count how many places the decimal was moved to the left.
4. Write the number in the form a×10ⁿ , where $n$ is positive.

3. Examples of Large Numbers in Standard Form

Example 1: Convert 450,000 to standard form

• Place decimal after the first digit: 4.5.
• Count the places moved: 5 places left.
• Standard form: 4.5×10^5

Example 2: Convert 72,000,000 to standard form

• Place decimal after 7.2.
• Count 7 places left.
• Standard form: 7.2×10^7

Example 3: Convert 6,030 to standard form

• Place decimal after 6.03.
• Count 3 places left.
• Standard form: 6.03×10^3

4. Converting Standard Form to Decimal Form

• A positive exponent means move the decimal to the right.

Example 4: Convert 3.1 × 10⁴ to decimal form

• Move decimal 4 places right: 31,000.

Example 5: Convert 9.85 × 10⁶ to decimal form

• Move decimal 6 places right: 9,850,000.
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For Small Numbers

1. Introduction

• When numbers are very small (less than 1), they can be written in standard form (scientific notation) using negative exponents of 10.
• Standard form helps in easy calculation and representation of small values.

2. General Formula

A small number is written in standard form as:

a×10-ⁿ

Where:

• $a$ is a number between 1 and 10 (including decimal values).
• 10^{-n}means the decimal moves n places to the left
• n is a positive integer.

3. Converting Small Numbers to Standard Form

Case 1: Decimal Present in the Number

Steps to Convert:

1. Identify the first non-zero digit.
2. Move the decimal point right until there is only one non-zero digit before the decimal.
3. Count how many places the decimal was moved. This gives the exponent (-n).
4. Write the number in the form a×10-ⁿ 

Example 1: Convert 0.00078 to standard form.

• The first non-zero digit is 7.
• Move the decimal 4 places right to get 7.8.
• Standard form: 7.8×10 ^−4

Example 2: Convert 0.0000432 to standard form.

• The first non-zero digit is 4.
• Move the decimal 5 places right to get 4.32.
• Standard form: 4.32×10^−5

Case 2: Decimal is at the End of the Number

• If a small number is written without a decimal (like 000456), place the decimal at the end and follow the same method.

Example 3: Convert 000024 to standard form.

• Write it as 0.00024.
• Move the decimal 4 places right to get 2.4.
• Standard form: 2.4×10^−4

4. Converting Standard Form Back to Decimal Form

• The negative exponent tells how many places to move the decimal left.

Example 4: Convert 5.9 × 10⁻³ to decimal form.

• Move decimal 3 places left: 0.0059

Example 5: Convert 3.07 × 10⁻⁶ to decimal form.

• Move decimal 6 places left: 0.00000307

5. Special Cases

Case 3: When the Number Already Starts with 1 to 9

• If the number is already between 1 and 10, it is not changed.

Example 6: Convert 3.5 to standard form.

• Already in correct form: 3.5×10^0

5. Summary