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1 The real numbers
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Australia
Year 9

1.04 Using scientific notation

Lesson
Worksheet
Practice
Lesson

Introduction

We will often use our calculator to evaluate expressions with scientific notation. However, knowing our index laws we can manipulate calculations that are relatively straightforward or estimate the size of answers for more complex calculations.

Use of scientific notation

A scientific calculator with the E X P button circled. Ask your teacher for more information.

Calculators will often display numbers in scientific notation but the format may vary between different models. A common variation from showing 2.95\times 10^8 is the display 2.95 \text{ E } 8 where the E is for exponent of 10. Most calculators will also have a button for entering numbers in scientific notation. This may look like \times 10^x or like the button EXP circled in blue in the picture. Look carefully at your calculator and ensure you are familiar with the display format and syntax for entering numbers in scientific notation.

Calculator buttons for 1 decimal point 5 E X P 9.

For example, to write 1.5\times 10^9 on this calculator, you would press the buttons shown on the left.

Examples

Example 1

Use your calculator to find the value of 82.97\times 7.1\times 10^4.

Express your answer using scientific notation.

Worked Solution
Create a strategy

Evaluate 82.97\times 7.1\times 10^4 using your calculator.

Apply the idea

Using the calculator, we get:

\displaystyle 82.97\times 7.1\times 10^4\displaystyle =\displaystyle 5.89087\times 10^6

Example 2

Use power laws to simplify 4\times 10^{-2}\times 9\times 10^9.

Express your answer using scientific notation.

Worked Solution
Create a strategy

Use the index law for multiplication and scientific notation.

Apply the idea
\displaystyle 4\times 10^{-2}\times 9\times 10^9\displaystyle =\displaystyle \left(4\times 9\right)\times 10^{-2+9}Use the multiplication law
\displaystyle =\displaystyle 36\times 10^7Evaluate the product 4\times9

Since 36 is greater than 10 we need to write it in scientific notation. 36 can be as expressed as 3.6\times 10^1.

\displaystyle 36\times 10^7\displaystyle =\displaystyle 3.6\times 10^1\times 10^{7}Write 36 in scientific notation
\displaystyle =\displaystyle 3.6\times 10^{8}Use the multiplication law

Example 3

A light year is defined as the distance that light can travel in one year. It is measured to be 9\,460\,730\,000\,000\,000 metres

a

Express a light year in metres using scientific notation.

Worked Solution
Create a strategy

Use the scientific notation form a \times 10^{n}, where a is a decimal number between 1 and 10 and n is a positive integer.

Apply the idea

To find the first part of our scientific notation we place the decimal point after the first non-zero number, soa=9.46073

9\,460\,730\,000\,000\,000 is 1\,000\,000\,000\,000\,000 or 10^{15} times bigger than 9.46073.9\,460\,730\,000\,000\,000=9.46073 \times 10^{15} \text{ metres}

b

Express a light year in kilometres using scientific notation.

Worked Solution
Create a strategy

Use the fact that 1 \text{ kilometre}= 1000 \text{ metres}, and divide our answer in part (a) by 1000.

Apply the idea

We need to divide the answer from part (a) by 1000=10^3.

\displaystyle 9.46073 \times 10^{15}\div 10^{3}\displaystyle =\displaystyle 9.46073 \times 10^{15-3}Use the division law
\displaystyle =\displaystyle 9.46073 \times 10^{12} \text{ km}Evaluate the subtraction
c

Express a light year in centimetres using scientific notation.

Worked Solution
Create a strategy

Use the fact that 1 \text{ metre}= 100 \text{ centimetres}. and multiply by 100.

Apply the idea

We need to multiply the answer from part (a) by 100=10^2.

\displaystyle 9.46073 \times 10^{15}\times 10^{2}\displaystyle =\displaystyle 9.46073 \times 10^{15+2}Use the multiplication law
\displaystyle =\displaystyle 9.46073 \times 10^{17} \text{ cm}Evaluate the addition

Example 4

The star Tindalos has a mass of 7.3\times 10^{50} kg and the star Cykranosh has a mass of 15.33\times 10^{53} kg.

By what factor is Cykranosh larger than Tindalos?

Express your answer as an integer using scientific notation.

Worked Solution
Create a strategy

Divide the mass of Cykranosh by the mass of Tindalos.

Apply the idea
\displaystyle \text{Factor}\displaystyle =\displaystyle \frac{15.33\times 10^{53}}{7.3\times 10^{50}}Divide the masses
\displaystyle =\displaystyle \frac{15.33}{7.3} \times 10^{53-50}Use division law
\displaystyle =\displaystyle \frac{15.33}{7.3} \times 10^{3}Evaluate the subtraction
\displaystyle =\displaystyle 2.1 \times 10^{3}Evaluate the fraction
Idea summary

We will often use our calculator to evaluate expressions with scientific notation. However, knowing our index laws can help us simplify or estimate our calculations.

Calculators will often display numbers in scientific notation but the format and button for entering numbers in scientific notation will vary between models.

Outcomes

ACMNA210

Express numbers in scientific notation

ACMMG219

Investigate very small and very large time scales and intervals

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