A basic representation of a number, such as 7,546, is designated as standard notation.

Expanded notation demonstrates the place value for each digit in a number by multiplying each digit by its place value and adding all the digits together.

For incredibly large or small numbers, such as 745,000,000,000,000 or 0.0000000095, scientific notation is used to show these numbers in more manageable forms.

Scientific notation converts the original number into a number with a ones’ place value followed by a few decimal place values.

This new number is then multiplied by 10 taken to a certain exponent.

- For numbers above the one’s place value, move the decimal to the left until the largest place value is converted to the one’s place value.
- The number of times you moved the decimal to the left is the 10’s power and it will be positive.
- For numbers below the one’s place value, move the decimal to the right until the first non‐zero digit is converted to the one’s place value.
- The number of times you moved the decimal to the right is the 10’s power and it will be negative.

To add and subtract using scientific notation, you have to keep in mind the concept of like terms. This is simple with the following example. Using the distributive property, you can factor out 104 and add the single digit values together.

**The same process works for subtraction: **

**However, when the powers of ten are different in the two terms**, you have to modify the problem before solving by writing both of the terms in terms of the same power of 10.

You can often do this by factoring out a 10 from one of the terms!

Notice at the end that you have to turn the answer back into scientific notation.

Multiplying and dividing using scientific notation is actually easier than addition and subtraction.

In **multiplication**, first you multiply the coefficients, then you multiply the powers of 10 using the Product Rule.

For example:

Once again, you see that you have to turn the answer back into scientific notation.

The process for **division** is similar. First, divide the coefficients, then divide the powers of 10 using the Quotient Rule.

**Answers to Practice Problems**

- Question 1
- 7(1,000) + 4(100) + 5(10) + 6(1) + 3(0.1)
- 1(1,000,000) + 9(100,000) + 2(10,000) + 3(1,000) + 4(100) + 3(10) + 2(10)
- 2(10) + 1(1) + 5(0.1) + 8(0.01) + 7(0.001)
- 5(10,000) + 5(1,000) + 3(100) + 4(10) + 5(1) + 9(0.1) + 9(0.01) + 9(0.001) + 9(0.0001)

- Question 2
- 7.89 x 10
^{8} - 6.734 x 10
^{10} - 2.04 x 10
^{-6} - 1.3077 x 10
^{-3}

- 7.89 x 10
- B
- D
- C
- C

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