Number & Algebra
Indices and Scientific Notation
Year 9
- ✓Apply the index laws to simplify algebraic expressions involving integer indices.
- ✓Convert numbers between standard form and scientific notation.
- ✓Perform calculations involving numbers expressed in scientific notation.
- ✓Understand the purpose and application of scientific notation for representing very large or very small numbers.
Key concepts
An index (also known as a power or exponent) indicates how many times a base number is multiplied by itself. For example, in the expression 5³, 5 is the base and 3 is the index, meaning 5 × 5 × 5. Understanding indices is fundamental for simplifying complex mathematical expressions.
Index laws are a set of rules used to simplify expressions involving powers. These laws apply when the bases are the same or when powers are raised to other powers.
When multiplying terms with the same base, add their indices.
When dividing terms with the same base, subtract the index of the denominator from the index of the numerator.
When raising a power to another power, multiply the indices.
The power of a product is equal to the product of each factor raised to that power.
The power of a quotient is equal to the quotient of the numerator and denominator each raised to that power.
Any non-zero base raised to the power of zero is equal to 1.
A base raised to a negative index is equal to the reciprocal of the base raised to the positive index.
Scientific notation is a standardised way of writing very large or very small numbers concisely. It expresses a number as a product of a number between 1 and 10 (inclusive of 1, exclusive of 10) and a power of 10. This makes it easier to read, compare, and perform calculations with such numbers.
Key facts to remember
- 1a^m × a^n = a^(m+n)
- 2a^m ÷ a^n = a^(m-n)
- 3(a^m)^n = a^(mn)
- 4(ab)^n = a^n b^n
- 5a^0 = 1 (for a ≠ 0)
- 6a^(-n) = 1/a^n (for a ≠ 0)
- 7Scientific notation is written as a × 10^n, where 1 ≤ |a| < 10 and n is an integer.
- 8A positive exponent in scientific notation indicates a large number; a negative exponent indicates a small number.
Worked examples
Example 1
Simplify the expression: (4x³y⁴)² × (2x⁻²y³)
Answer
32x⁴y¹¹
Remember to apply the power to all factors inside the brackets.
Example 2
a) Express 0.00000078 in scientific notation. b) Express 6.02 × 10⁵ in standard form.
Answer
a) 7.8 × 10⁻⁷\nb) 602 000
A negative exponent means a small number (less than 1), and a positive exponent means a large number (greater than 10).
Example 3
Calculate (3.5 × 10⁴) × (2 × 10⁶). Express your answer in scientific notation.
Answer
7 × 10¹⁰
Always ensure the numerical part of your scientific notation answer is between 1 and 10.
Common mistakes
- ✗Adding indices when the bases are different (e.g., x² × y³ ≠ (xy)⁵).
- ✗Incorrectly applying the zero index law (e.g., writing 5x⁰ = 0 instead of 5 × 1 = 5).
- ✗Confusing negative indices with negative numbers (e.g., 3⁻¹ ≠ -3; it's 1/3).
- ✗Not ensuring the 'a' part of scientific notation (a × 10^n) is between 1 and 10 (e.g., writing 25 × 10³ instead of 2.5 × 10⁴).
- ✗Incorrectly moving the decimal point when converting to/from scientific notation, especially with negative powers of 10.
Exam tips
- ★Always show your working step-by-step, especially when applying multiple index laws, to earn partial marks.
- ★Double-check the sign of the index when dealing with negative indices or division, as this is a common source of error.
- ★When converting to scientific notation, verify that the power of 10 is positive for large numbers and negative for small numbers.
- ★Ensure your final answer in scientific notation has the first part (the 'a' value) strictly between 1 and 10 (1 ≤ |a| < 10).
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