Explainer: Numbers Operations in Scientific Notation

In this explainer, we will learn how to perform arithmetic operations with numbers expressed in scientific notation.

Before looking at how to perform calculations with numbers in scientific notation, let us quickly recap how to convert numbers between standard form and scientific notation.

Consider the number 156,000,000,000. A number is considered to be in scientific notation if it is in the form π‘˜Γ—10𝑛, where 1β‰€π‘˜<10 and 𝑛 is an integer. Therefore, we need to establish how many times we would need to divide 156,000,000,000 by 10 to reduce it in magnitude to a number between 1 and 10. This would be 11 times and the number would be 1.56. Hence, if we multiplied 1.56 by 10 eleven times, we would get back to our original number. So, 156,000,000,000 in scientific notation is 1.56Γ—1011.

Similarly, consider the number 0.000000000324. In order to convert this number into scientific notation we first need to establish how many times we would need to multiply it by 10 to increase its magnitude to a number between 1 and 10. This would be 10 times, and we would get the number 3.24. If we then divided 3.24 by 10 ten times, we would get back to our original number. Hence, in scientific notation we would write 0.000000000324 as 3.24Γ—10βˆ’10.

Notice here that we have used negative indices to avoid writing a divide sign.

Now that we have recapped how to convert numbers into scientific notation, let us look at how to perform calculations with them in this form. In order to do this, we need to recap some key mathematical concepts. Firstly, in the real number system, numbers are commutative under multiplication. In other words π‘ŽΓ—π‘=π‘Γ—π‘Ž.

Secondly, if we multiply two numbers with the same base, we can just add the exponents: π‘₯π‘ŽΓ—π‘₯𝑏=π‘₯π‘Ž+𝑏, and if we divide two numbers with the same base, we subtract the exponents: π‘₯π‘ŽΓ·π‘₯𝑏=π‘₯π‘Žβˆ’π‘.

In order to see the relevance of these two rules, let us look at an example. Consider the multiplication 3Γ—107Γ—2Γ—108.

To evaluate this multiplication we can first apply the commutative property to rewrite the multiplication: 3Γ—2Γ—107Γ—108.

We can then evaluate the multiplication 3Γ—2 and simplify the powers of 10 to get 6Γ—1015, which is an expression in scientific notation. If we wanted to write this expression in standard form, we would write 6 multiplied by 10 fifteen times: 6,000,000,000,000,000.

Now, this is a fairly introductory level example where the answer is straight away presented in scientific notation. Let us look now at an example where this is not the case.

Example 1: Performing Calculations with Numbers in Scientific Notation

Express ο€Ή9Γ—10611Γ—104 in scientific notation.

Answer

Firstly, we use the commutative property to rewrite the expression as 9Γ—11Γ—106Γ—104.

If we evaluates the multiplication and then simplify the powers of 10 we get 99Γ—1010.

This, however, is not yet in scientific notation as 99 is not a number between 1 and 10. If we rewrite the expression so that the 99 is written in scientific notation we get 9.9Γ—10Γ—1010, which simplifies to 9.9Γ—1011.

Let us now look at example that contains decimals.

Example 2: Performing Calculations with Numbers in Scientific Notation

Calculate 1.2Γ—103Γ—2.4Γ—102 giving your answer in scientific notation.

Answer

There are a couple of approaches for answering this question and we will demonstrate both of these. In the first method, we start by rearranging the expression using the commutative property to be 1.2Γ—2.4Γ—103Γ—102.

If we then multiply 1.2 and 2.4, we get 2.88. If we then group the powers of 10, we get 105. Combining these two calculations, we then find that our solution is 2.88Γ—105. Now, if you find multiplying decimals quite difficult, you may want to follow a different approach which is our second method. Starting with the original expression, we can β€œborrow a 10” from each power of 10 to remove the decimals: 1.2Γ—10Γ—102Γ—2.4Γ—10Γ—101.

This simplifies to 12Γ—102Γ—24Γ—101.

We can now rearrange the expression using the commutative property to 12Γ—24Γ—102Γ—101, which, when we complete the multiplication and group the powers, gives 288Γ—103.

The problem now is that this expression is not in scientific notation, so our final step is to rewrite 288 in scientific notation and then simplify: 2.88Γ—102Γ—103 simplifies to 2.88Γ—105.

Apparently here the second method is longer but it may help you understand the method more clearly.

Now let us look at questions involving division. Here, we take a very similar approach but have to be a little careful when we rearrange the expressions. We will demonstrate the method with another example.

Example 3: Performing Calculations with Numbers in Scientific Notation Involving Division

Calculate ο€Ή8.2Γ—104÷4.1Γ—102, giving your answer in scientific notation.

Answer

Firstly, we can rewrite the expression as 8.2Γ—1044.1Γ—102; this can then be rewritten as a multiplication of two fractions: 8.24.1Γ—104102.

Simplifying each fraction separately, we then get 2Γ—102.

Note, for an intermediary step, we could have rewritten the two fractions as (8.2Γ·4.1)Γ—ο€Ή104Γ·102 and simplified from here.

Let us now look at an example of questions involving negative exponents. Here, we have to be particularly careful with operations with negative numbers.

Example 4: Performing Calculations with Numbers in Scientific Notation Involving Division

Find the value of ο€Ή4.4Γ—10βˆ’2÷1.1Γ—10βˆ’6, giving your answer in standard form.

Answer

Firstly, we can rewrite the expression as 4.4Γ—10βˆ’21.1Γ—10βˆ’6; this can then be rewritten as a multiplication of two fractions: 4.41.1Γ—10βˆ’210βˆ’6.

The first fraction simplifies to 4. The second fraction can be rewritten as 10βˆ’2Γ·10βˆ’6.

This is then the same as 10βˆ’2βˆ’(βˆ’6), which simplifies to 104.

Therefore, the final answer is 4Γ—104.

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