GCSE Maths Practice: standard-form

Question 6 of 10

This higher-tier problem applies standard form to scientific contexts involving small decimal values and powers of ten.

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\( \begin{array}{l}\text{A virus has radius } 0.000000732 \text{ m.} \\ \text{It is magnified } 4 \times 10^3 \text{ times.} \\ \text{Find its apparent size in standard form.}\end{array} \)

Choose one option:

Handle the coefficient and exponent separately. Multiply coefficients directly and add exponents using the law of indices.

Working with Very Small Numbers in Standard Form

Standard form is essential when dealing with numbers that are extremely small or large. In science and engineering, many measurements—such as the size of atoms, wavelengths of light, or virus diameters—are written as decimals with several leading zeros. Converting these into powers of ten simplifies calculations and allows patterns to be compared easily.

The Concept of Standard Form

A number in standard form has the structure \( a \times 10^n \), where \( 1 \leq a < 10 \) and \( n \) is an integer. The exponent \( n \) shows how many places the decimal point has been moved. If the number is less than one, \( n \) is negative, indicating division by powers of ten.

For example, \( 0.000000732 = 7.32 \times 10^{-7} \) because the decimal moves seven places to the right to make \( 7.32 \).

Combining Numbers in Standard Form

Once numbers are in standard form, you can apply the laws of indices to multiply or divide them efficiently:

  • \( (a \times 10^m) \times (b \times 10^n) = (a \times b) \times 10^{m+n} \)
  • \( \dfrac{a \times 10^m}{b \times 10^n} = \left(\dfrac{a}{b}\right) \times 10^{m-n} \)

Worked Example 1: Particle Size Calculation

A virus particle has a radius of \( 0.000000732 \text{ m} \). A microscope magnifies this particle \( 4 \times 10^3 \) times. Find the apparent size after magnification.

Step 1: Convert to standard form.
\( 0.000000732 = 7.32 \times 10^{-7} \)

Step 2: Multiply by the magnification.
\( (7.32 \times 10^{-7}) \times (4 \times 10^3) = (7.32 \times 4) \times 10^{-7+3} \)

Step 3: Simplify.
\( 29.28 \times 10^{-4} = 2.93 \times 10^{-3} \)

The apparent size is \( 2.93 \times 10^{-3} \text{ m} \).

Worked Example 2: Scientific Comparison

The wavelength of red light is approximately \( 0.00000065 \text{ m} \). Express this in standard form.

\( 0.00000065 = 6.5 \times 10^{-7} \). This means the wavelength is 650 nanometres (nm), a common scientific unit used for light.

Common Mistakes

  • Using positive instead of negative exponents: Remember, small numbers below one always use negative powers.
  • Forgetting to normalise: The coefficient must always be between 1 and 10. \( 29.28 \times 10^{-4} \) must be rewritten as \( 2.93 \times 10^{-3} \).
  • Incorrectly moving the decimal: Count the zeros carefully. Each shift to the right adds 1 to the negative power.

Real-World Relevance

Standard form is vital in modern science. Physicists describe atomic radii (around \( 10^{-10} \text{ m} \)), astronomers record planetary distances (around \( 10^{11} \text{ m} \)), and engineers model circuits involving nano-scale components. Being able to manipulate small and large values precisely is essential for clear communication and accurate results.

FAQs

Q1: Why is the power negative for small numbers?
A: Because moving the decimal to the right indicates division by powers of ten. Each move represents one factor of \( 10^{-1} \).

Q2: What happens if the coefficient exceeds 10 after multiplication?
A: Move the decimal point left until it is between 1 and 10, increasing the power of ten accordingly.

Q3: Are negative powers ever used for large numbers?
A: No. Negative powers are only used for values less than one.

Study Tip

When working with very small numbers, write the zeros first and count them carefully before moving the decimal. Always double-check that the final coefficient lies between 1 and 10 and that your power of ten matches the direction of movement. This attention to detail is crucial in GCSE Maths Higher topics involving standard form and scientific notation.