Indices, Standard Form and Surds

Prime Factor Decomposition

Highest Common Factor and Lowest Common Multiple

Rules of Indices

Standard Index Form

Adding and Subtracting in Standard Form

Calculations with Standard Form

Negative Powers / Reciprocals

Fractional Powers

Simplifying Surds

Calculating with Surds

Rationalising the Denominator

Rules of Indices

Writing Numbers in Standard Form

Negative and Fractional Powers

Prerequisite Knowledge

• Apply the four operations, including formal written methods, to integers
• Use and interpret algebraic notation
• Count backwards through zero to include negative numbers
• Use negative numbers in context, and calculate intervals across zero

Success Criteria

• Use the concepts and vocabulary of highest common factor, lowest common multiple, prime factorisation, including using product notation and the unique factorisation theorem
• Calculate with roots, and with integer and fractional indices
• Calculate with and interpret standard form A x 10n, where 1 ≤ A < 10 and n is an integer.
• Simplify and manipulate algebraic expressions
• Simplifying expressions involving sums, products and powers, including the laws of indices
• Calculate exactly with surds
• Simplify surd expressions involving squares and rationalise denominators

Key Concepts

• To decompose integers into their prime factors students may need to review the definition of a prime.
• A base raised to a power of zero has a value of one.
• Students need to have a secure understanding in the difference between a highest common factor and lowest common multiple.
• Standard index form is a way of writing and calculating with very large and small numbers. A secure understanding of place value is needed to access this.
• Surds are square roots that cannot exactly in fraction form.
• Students need to generalise the rules of indices.

Common Misconceptions

• One is not a prime number since it only has one factor.
• x² is often incorrectly taken with 2x.
• Students often have difficulty when dealing with negative powers. For instance, 1.2 × 10^-2 they assume,  to have a value of -120.
• Multiplying out brackets involving surds is often attempted incorrectly.
• √(5²) is often confused with 2√5