Introduction

Standard form is a way of writing numbers using powers of 10. It is especially useful for representing very large or very small numbers in a concise and understandable manner. In standard form, a number is expressed as the product of a number between 1 and 10 and a power of 10.

General Form of Standard Form:
A number in standard form is written as:
A \times 10^b
where:
* A is a number between 1 and 10
* b is an integer (positive, negative, or zero)
Examples
* 2345 in standard form is 2.345 \times 10^3.
* 0.00005 in standard form is 5 \times 10^{-5}.

Converting Numbers to Standard Form

1. Identify the decimal point: Locate the decimal point in the number.
2. Move the decimal point: Move the decimal point to the right of the first non-zero digit.
3. Count the number of places moved: This number will be the exponent of 10.
4. Write the number in standard form: Multiply the number with the decimal point in the correct position by 10 raised to the power you found in step 3.

Example

Convert 567800 to standard form.

1. The decimal point is after the 0.
2. Move the decimal point to the right of the 5: 5.678
3. We moved the decimal point 5 places to the right, so the exponent is 5.
4. Therefore, 567800 in standard form is 5.678 \times 10^5.

Converting Numbers from Standard Form

1. Identify the exponent of 10: This indicates the number of places to move the decimal point.
2. Move the decimal point: If the exponent is positive, move the decimal point to the right. If the exponent is negative, move the decimal point to the left.

3. Add zeros if necessary: Add zeros to the end of the number if needed to complete the decimal places.

Example

Convert 3.45 \times 10^{-2} to standard form.

1. The exponent is -2, so we move the decimal point 2 places to the left.
2. 3.45 becomes 0.0345.

Therefore, 3.45 \times 10^{-2} in standard form is 0.0345.

Exercises

1. Convert 234,000 to standard form.
2. Convert 0.00000789 to standard form.
3. Write 5.67 \times 10^4 in standard form.
4. Write 8.9 \times 10^{-3} in standard form.
5. Convert 123,456,789 to standard form.