GCSE Maths Practice: standard-form

Question 5 of 10

This Higher-tier question applies standard form to represent very large scientific quantities such as planetary distances.

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\( \begin{array}{l}\text{A newly discovered planet orbits its star at } \\ 98700000000 \text{ m. Express this distance in standard form.}\end{array} \)

Choose one option:

For large numbers, shift the decimal left and use a positive power. Check that the coefficient remains between 1 and 10.

Converting Very Large Numbers to Standard Form

Large numbers appear frequently in GCSE Maths, especially in scientific contexts such as astronomy, physics, and economics. Writing them in standard form simplifies reading, comparing, and calculating values that would otherwise be cumbersome to handle. For example, the population of Earth or the distance between planets often contains many zeros, and standard form allows those quantities to be written more clearly.

The Principle

In standard form, a number is written as \( a \times 10^n \), where \( 1 \leq a < 10 \) and \( n \) is an integer. For large numbers, \( n \) is positive and represents how many places the decimal point moves to the left. Each leftward move corresponds to dividing by 10 once, while each rightward move (for small numbers) corresponds to multiplying by 10 once.

Worked Example 1

Convert \( 98700000000 \) into standard form:

  1. Identify where the decimal point is — at the end of the number.
  2. Move it left until only one non-zero digit remains before the point: \( 9.87 \).
  3. Count the moves: 10 places left.
  4. Write the number as \( 9.87 \times 10^{10} \).

This means the number is approximately 9.87 multiplied by ten to the power of ten, or 98,700,000,000 in full form.

Worked Example 2: Scientific Application

The average distance between Earth and the Sun is about 149600000000 m. To express this in standard form, move the decimal 11 places left:

\( 1.496 \times 10^{11} \).
This compact form allows scientists to record distances between planets and stars without writing long sequences of zeros.

Combining with Calculations

Once a value is in standard form, it can easily be multiplied or divided with others using index laws. For example:

\[ (3.2 \times 10^5) \times (9.87 \times 10^{10}) = (3.2 \times 9.87) \times 10^{5+10} = 31.6 \times 10^{15} = 3.16 \times 10^{16}. \]

This process saves time and reduces the chance of counting zeros incorrectly.

Common Mistakes

  • Incorrect exponent count: Always count the number of decimal shifts carefully. Double-check by reconstructing the number from your answer.
  • Coefficient not between 1 and 10: Remember to adjust the decimal position so that only one non-zero digit appears before the point.
  • Wrong sign for the exponent: Large numbers use positive exponents, small numbers use negative ones.

Real-World Relevance

Standard form is essential in areas such as astronomy, where distances are measured in millions or billions of kilometres. It also appears in computing (data sizes), engineering (forces and energies), and finance (national debts or GDPs). In all these cases, expressing data in powers of ten improves communication and accuracy.

FAQs

Q1: How do I know whether the power should be positive or negative?
A: If the original number is greater than 1, the power is positive. If it is smaller than 1, the power is negative.

Q2: Does rounding affect the power?
A: No. Rounding only changes the coefficient. The power depends solely on how far the decimal moves.

Q3: What happens if my coefficient ends up greater than 10?
A: Move the decimal one more place left and increase the exponent by 1.

Study Tip

When converting large numbers, group digits in threes to count decimal shifts faster (e.g., 987 000 000 00 → ten moves). After writing the number in standard form, quickly check that expanding it again reproduces the original value. This habit ensures accuracy and speed in GCSE Maths Higher questions on Standard Form.