Open Math›Year 10›Measurement precision›Scientific notation applications

Scientific notation applications

Convert between standard form and scientific notation, perform multiplication and division in scientific notation, and solve real-world problems involving very large and very small numbers.

Worked examples

Converting a small number to scientific notation

Straightforward

Problem

Write

0.000307

in scientific notation.

Move the decimal point right until the number is between 1 and 10.

Moving the decimal point 4 places to the right:

0.000307 \to 3.07

Determine the power of 10. Since the original number is less than 1, the exponent is negative. The number of places moved gives the magnitude of the exponent.

Moved 4 places to the right

\Rightarrow

exponent is

-4

Write the number in scientific notation and verify.

0.000307 = 3.07 \times 10^{-4}

Check:

3.07 \times 10^{-4} = 3.07 \div 10\,000 = 0.000307

✓

Answer

3.07 \times 10^{-4}

Multiplying numbers in scientific notation

Moderate

Problem

Calculate

(6.2 \times 10^5) \times (3 \times 10^{-2})

and express the answer in scientific notation.

Multiply the coefficients.

6.2 \times 3 = 18.6

Multiply the powers of 10 by adding the exponents.

10^5 \times 10^{-2} = 10^{5 + (-2)} = 10^3

Combine the results.

18.6 \times 10^3

Adjust so the coefficient is between 1 and 10.

18.6 = 1.86 \times 10^1

, so

18.6 \times 10^3 = 1.86 \times 10^{1+3} = 1.86 \times 10^4

Answer

1.86 \times 10^4

Solving a real-world problem using scientific notation

Challenging

Problem

A spacecraft travels at

2.4 \times 10^4

km/h. It needs to cover a distance of

1.8 \times 10^9

km to reach a space station. How many hours will the journey take? Express the answer in scientific notation.

Write the formula for time.

\text{time} = \frac{\text{distance}}{\text{speed}} = \frac{1.8 \times 10^9}{2.4 \times 10^4}

Divide the coefficients.

1.8 \div 2.4 = 0.75

Divide the powers of 10 by subtracting the exponents.

10^9 \div 10^4 = 10^{9-4} = 10^5

Combine and adjust so the coefficient is between 1 and 10.

0.75 \times 10^5 = 7.5 \times 10^{-1} \times 10^5 = 7.5 \times 10^4

Answer

7.5 \times 10^4

hours

Practise

Q1·Straightforward

Which of the following is

4\,700\,000

written in scientific notation?

$4.7 \times 10^5$$4.7 \times 10^6$$47 \times 10^5$$0.47 \times 10^7$

Explanation

Move the decimal point 6 places to the left to get

4.7

. Since the original number is large (greater than 1), the exponent is positive:

4.7 \times 10^6

. The other options are not in correct scientific notation — the coefficient must be between 1 and 10.

Q2·Straightforward

Which of the following is

0.000052

written in scientific notation?

$5.2 \times 10^{-4}$$5.2 \times 10^{-5}$$52 \times 10^{-6}$$5.2 \times 10^{5}$

Explanation

Move the decimal point 5 places to the right to get

5.2

. Since the original number is small (less than 1), the exponent is negative:

5.2 \times 10^{-5}

. Check:

5.2 \times 10^{-5} = 5.2 \div 100\,000 = 0.000052

✓

Q3·Straightforward

Write

6.3 \times 10^5

as a basic numeral.

Explanation

6.3 \times 10^5 = 6.3 \times 100\,000 = 630\,000

. Moving the decimal point in

6.3

five places to the right gives

630\,000

Q4·Straightforward

Which number is larger:

7.2 \times 10^5

9.8 \times 10^4

$7.2 \times 10^5$, because it has a higher power of 10$9.8 \times 10^4$, because 9.8 is greater than 7.2They are equalCannot be determined without a calculator

Explanation

7.2 \times 10^5 = 720\,000

and

9.8 \times 10^4 = 98\,000

. When comparing numbers in scientific notation, compare the exponents first. Since

10^5 > 10^4

, the number

7.2 \times 10^5

is larger — even though

9.8 > 7.2

as coefficients.

Q5·Moderate

Evaluate

(3 \times 10^4) \times (2 \times 10^3)

and express the answer in scientific notation.

$6 \times 10^7$$6 \times 10^{12}$$5 \times 10^7$$6 \times 10^{43}$

Explanation

Multiply the coefficients:

3 \times 2 = 6

. Add the exponents:

10^4 \times 10^3 = 10^{4+3} = 10^7

. Answer:

6 \times 10^7

Q6·Moderate

Evaluate

\dfrac{8.4 \times 10^9}{2.1 \times 10^3}

and express the answer in scientific notation.

$4 \times 10^3$$4 \times 10^6$$4 \times 10^{12}$$6.3 \times 10^6$

Explanation

Divide the coefficients:

8.4 \div 2.1 = 4

. Subtract the exponents:

10^9 \div 10^3 = 10^{9-3} = 10^6

. Answer:

4 \times 10^6

Q7·Moderate

Calculate

(4.5 \times 10^6) \times (2 \times 10^{-2})

and write the answer as a basic numeral.

Explanation

Multiply coefficients:

4.5 \times 2 = 9

. Add exponents:

10^6 \times 10^{-2} = 10^{6+(-2)} = 10^4

. Result:

9 \times 10^4 = 90\,000

Q8·Moderate

Calculate

\dfrac{7.2 \times 10^8}{9 \times 10^4}

and write the answer as a basic numeral.

Explanation

Divide coefficients:

7.2 \div 9 = 0.8

. Subtract exponents:

10^8 \div 10^4 = 10^{8-4} = 10^4

. Result:

0.8 \times 10^4 = 8 \times 10^3 = 8000

Q9·Challenging

A bacterium is

2.5 \times 10^{-6}

m long. A virus is

5 \times 10^{-8}

m long. How many times longer is the bacterium than the virus?

Explanation

\frac{2.5 \times 10^{-6}}{5 \times 10^{-8}} = \frac{2.5}{5} \times 10^{-6-(-8)} = 0.5 \times 10^{2} = 5 \times 10^{1} = 50

The bacterium is 50 times longer than the virus.

Q10·Challenging

The diameter of the Sun is approximately

1.39 \times 10^9

m. The diameter of the Earth is approximately

1.27 \times 10^7

m. What is the difference in their diameters, in metres, written in scientific notation (to 3 significant figures)?

$1.38 \times 10^9$ m$1.26 \times 10^9$ m$1.52 \times 10^9$ m$1.38 \times 10^{10}$ m

Explanation

Express both in terms of

10^9

1.39 \times 10^9 - 0.0127 \times 10^9 = (1.39 - 0.0127) \times 10^9 = 1.3773 \times 10^9

Rounded to 3 significant figures:

1.38 \times 10^9

Q11·Challenging

Australia has approximately

2.6 \times 10^7

people. Each person uses approximately

4.0 \times 10^2

litres of water per day. What is the total daily water usage in Australia, in litres, written in scientific notation?

$1.04 \times 10^{10}$ litres$1.04 \times 10^9$ litres$10.4 \times 10^9$ litres$6.6 \times 10^9$ litres

Explanation

Multiply coefficients:

2.6 \times 4.0 = 10.4

. Add exponents:

10^7 \times 10^2 = 10^9

. This gives

10.4 \times 10^9

, but the coefficient must be between 1 and 10. Rewrite:

10.4 \times 10^9 = 1.04 \times 10^{10}

litres.

Q12·Challenging

Light travels at approximately

3 \times 10^8

metres per second. The distance from the Earth to the Sun is approximately

1.5 \times 10^{11}

metres. How many seconds does it take light to travel from the Sun to Earth?

Explanation

\text{time} = \frac{1.5 \times 10^{11}}{3 \times 10^8} = \frac{1.5}{3} \times 10^{11-8} = 0.5 \times 10^3 = 5 \times 10^2 = 500 \text{ seconds}

Light takes 500 seconds (about 8 minutes 20 seconds) to reach Earth from the Sun.