# Scientific Notation

# Addition, Subtraction, Division and Multiplication

Scientific notation is a standard form of writing very small and very big numbers so that they can easily be used for computations and other calculations.

For example: 300 is written as 3 × 10^{2} in scientific notation.

A number in scientific notation is of the form *a* × 10* ^{n}*, where

*a*is a number between 1 and 9, inclusive and

*n*can be a positive or a negative integer. The decimal point of a number is moved till we get a number from 1 to 9 and value of

*n*is the number of places the decimal is moved.

If we move the decimal point towards the left, then the exponent will be positive.

If we move the decimal point towards the right, then the exponent will be negative.

See the examples below:

*Example*: Write 8,380,000 in scientific notation.

8,380,000 is a whole number so decimal point will be at the end of the number, so here *a* = 8.38

Now *n* is the number of places the decimal is moved. Here we moved the decimal point towards the left, so exponent will be positive. So, *n* = 6.

So, 8,380,000 in scientific notation is written as **8.38 ****×**** 10 ^{6}**

*Example*: Write 0.00061 in scientific notation.

Here we move the decimal to the right so that we get a number from 1 to 9. So, *a* = 6.1

*n* is the number of places the decimal is moved. Here we moved the decimal point towards the right, so exponent will be negative. So, *n* = -4.

So, 0.00061 in scientific notation is written as **6.1** **×**** 10 ^{-4}**

**Addition and Subtraction in Scientific Notation**

We know how to write numbers in scientific notation, which means numbers have the same base that is 10. Next, let us go through the steps to be followed for adding or subtracting:

*Step 1*: Make the exponents equal to each other for which we should make the smaller exponent equal to the larger one by dividing with appropriate power of 10.

*Step 2*: Add or subtract the new coefficients.

*Step 3*: Write the answer in scientific notation, if it is not in scientific notation after adding or subtracting.

*Example*: Subtract 5.66 × 10^{1} – 3 × 10^{-2}

Here -2 is the smaller exponent, so multiply 3 × 10^{-2} by 10^{3} to make it 10^{1 }and divide by 10^{3},

So, 3 × 10^{-2} = = = = =

Now 5.66 × 10^{1} – 3 × 10^{-2} = 5.66 × 10^{1} – 0.003 × 10^{1} = (5.66 – 0.003) × 10^{1} = **5.657 ****×**** 10 ^{1}**

**Multiplication and Division in Scientific Notation**

To multiply or divide the numbers in scientific notation, we multiply or divide their coefficients and add their exponents. If after adding or subtracting the coefficients, the answer is not in scientific notation, then convert it to scientific notation.

*Step 1*: Multiply or divide the coefficients.

*Step 2*: Add or subtract the exponents.

*Step 3*: Convert the answer in scientific notation.

*Example*: Multiply (3.1 × 10^{-1})(0.4 × 10^{2})

*Step 1*: Multiply 3.1 and 0.4 that is 3.1 × 0.4 = 1.24

*Step 2*: Add the exponents, 10^{-1} × 10^{2} = 10^{-1+2} = 10^{1}

*Step 3*: (3.1 × 10^{-1})(0.4 × 10^{2}) = 1.24 × 10^{1}

**CHECK POINT**

Simplify and write each of the following in scientific notation.

- 0.000251
- 0.792 × 10
^{5} - (6.3 × 10
^{-2})(6.2 × 10^{2}) - (5.4 × 10
^{-1}) ÷ (2.7 × 10^{1}) - (3 × 10
^{-2})(9 × 10^{4}) - (3.6 × 10
^{3}) ÷ (0.6 × 10^{3})

##### Answer key

- 2.51 × 10
^{-4} - 7.92 × 10
^{4} - 3.906 × 10
^{1} - 2 × 10
^{-2} - 2.7 × 10
^{3} - 6

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