AQA GCSE Maths

Rationalising denominators

AQA GCSE Maths revision on Rationalising denominators. Aligned to the AQA GCSE Mathematics 8300 specification. This bank has 16 practice questions on this topic.

Sample questions (3 of 16)

  1. Question 1

    When rationalising the denominator of 1/sqrt(5), why do we multiply the numerator and denominator by sqrt(5)?

    • A) To change the value of the fraction
    • B) To create an equivalent fraction with a rational denominator
    • C) To simplify the numerator into a whole number
    • D) Because sqrt(5) is a prime number
    Show answer

    Answer: To create an equivalent fraction with a rational denominator

    Multiplying a fraction by a form of 1, such as sqrt(5)/sqrt(5), does not change the overall value of the expression. The goal of rationalising is specifically to transform the denominator into an integer, which makes the expression easier to work with algebraically.

  2. Question 2

    A student attempts to rationalise 1/(sqrt(3) + 1) by multiplying the numerator and denominator by (sqrt(3) + 1). Why is this strategy incorrect?

    • A) It results in a denominator that still contains a surd
    • B) It changes the overall value of the fraction
    • C) It creates a fraction that is no longer in its simplest form
    • D) The numerator should be multiplied by (sqrt(3) - 1) instead
    Show answer

    Answer: It results in a denominator that still contains a surd

    When dealing with a binomial denominator, you must multiply by the conjugate to utilise the difference of two squares identity. Multiplying by the original binomial results in a cross-product term that still contains the square root, failing to achieve a rational denominator.

  3. Question 3

    What is the primary algebraic identity used when rationalising a denominator containing a binomial surd like (a + sqrt(b))?

    • A) Difference of two squares
    • B) Completing the square
    • C) Quadratic formula
    • D) Expanding binomials
    Show answer

    Answer: Difference of two squares

    The difference of two squares identity is essential because squaring a surd turns it into a rational number. By multiplying the denominator by its conjugate, the middle terms cancel out, leaving only the squares of the individual components.

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