GCSE Maths Practice: standard-form

Question 8 of 10

This higher-tier question tests your ability to add and subtract numbers in standard form when exponents differ.

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\( \begin{array}{l}\text{Calculate } (5.8 \times 10^5) + (7.1 \times 10^4) - (3.2 \times 10^3), \\ \text{giving your answer in standard form to 3 s.f.}\end{array} \)

Choose one option:

Convert smaller powers to match the largest exponent, then add or subtract coefficients carefully before writing the result in standard form.

Adding and Subtracting Numbers in Standard Form

In GCSE Maths, students often work with numbers in standard form, also called scientific notation. This format is especially helpful when handling very large or very small numbers. It is widely used in science, engineering, and finance, where quantities can vary by several powers of ten. The key is learning how to combine such values efficiently and accurately.

Understanding the Principle

Standard form expresses a number as \( a \times 10^n \), where \( 1 \leq a < 10 \) and \( n \) is an integer. To add or subtract numbers in standard form, both numbers must share the same power of ten. Only the coefficients (the \( a \) values) can be added or subtracted directly once the exponents are equal.

If the powers differ, adjust one of the numbers by moving the decimal in its coefficient and changing the exponent accordingly. After both numbers use the same exponent, you can safely combine their coefficients and then rewrite the result in standard form if needed.

Step-by-Step Strategy

  1. Identify each term and note its power of ten.
  2. Convert smaller or larger exponents to match the largest one.
  3. Add or subtract the coefficients.
  4. Write the new value in standard form so that \( 1 \leq a < 10 \).
  5. Round the final answer to the required number of significant figures.

Worked Example 1: Mixed Powers

Example: \((4.2 \times 10^3) + (8.5 \times 10^4)\)
Convert \(4.2 \times 10^3\) to \(0.42 \times 10^4\).
Then, \((0.42 + 8.5) \times 10^4 = 8.92 \times 10^4\).
The result is already in standard form.

Worked Example 2: Positive and Negative Powers

Example: \((3.1 \times 10^{-2}) + (5.4 \times 10^{-3})\)
Match powers by rewriting \(5.4 \times 10^{-3} = 0.54 \times 10^{-2}\).
Add coefficients: \((3.1 + 0.54) \times 10^{-2} = 3.64 \times 10^{-2}\).

Common Mistakes

  • Adding exponents directly: The power of ten stays the same once aligned — only the coefficients change.
  • Skipping coefficient adjustment: Always rewrite the number with a consistent exponent before adding or subtracting.
  • Incorrect final form: After the operation, ensure the coefficient lies between 1 and 10; adjust the power if necessary.

Real-World Applications

Understanding standard form addition and subtraction is crucial for many fields:

  • Science: Astronomers sum vast distances, and chemists add small atomic-scale values.
  • Engineering: Used in measurements of voltage, current, and resistance.
  • Finance: Large-scale figures like national budgets or corporate revenues are often expressed using powers of ten.

FAQs

Q1: What if two numbers already share the same exponent?
A: Simply add or subtract their coefficients — no rewriting is needed.

Q2: Why is rounding important?
A: It reflects the level of precision in real-world measurements or data.

Q3: Can coefficients be negative?
A: Yes, negative numbers follow the same rules, just ensure correct sign handling when combining.

Study Tip

When practising, always write each conversion step neatly. Check that the power of ten remains consistent before and after adding or subtracting. With consistent use, this process becomes quick and intuitive — a core skill for the Standard Form and Indices section of GCSE Maths.