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Surds (Edexcel IGCSE Maths A)

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  • What is a surd?

    A surd is a square root of a non-square positive integer.

    It is an irrational number.

    Examples include square root of 7 comma space square root of 12 comma square root of 99.

  • Simplify square root of a cross times square root of b.

    square root of a cross times square root of b simplifies to square root of a cross times b end root.

  • Simplify fraction numerator square root of a over denominator square root of b end fraction.

    fraction numerator square root of a over denominator square root of b end fraction simplifies to square root of a over b end root.

  • Simplify open parentheses square root of a close parentheses squared.

    open parentheses square root of a close parentheses squared simplifies to a.

  • True or False?

    square root of 2 plus square root of 3 simplifies to square root of 5.

    False.

    square root of 2 plus square root of 3 does not simplify to square root of 5.

    The surds need to be of the same type (e.g. all multiple of square root of 2 ) to be able to add or subtract.

  • How can you simplify a surd such as square root of 8?

    To simplify a surd, write the number as a product involving a square number.

    Split the surd into the product of two surds, one of which should just become an integer.

    E.g. square root of 8 equals square root of 4 cross times 2 end root equals square root of 4 cross times square root of 2 equals 2 square root of 2.

  • How do you add or subtract two surds?

    E.g. square root of 8 plus square root of 18.

    To add or subtract two surds, simplify both surds so that they are both multiples of the same surd.

    You can then add or subtract by collecting like terms.

    For example, square root of 8 plus square root of 18 equals 2 square root of 2 plus 3 square root of 2 equals 5 square root of 2

  • How do you expand two brackets with expressions containing surds?

    E.g. open parentheses 2 plus square root of 3 close parentheses open parentheses 4 plus square root of 2 close parentheses

    To expand two brackets with expressions containing surds, you multiply each term in one bracket by each term in the other bracket.

    E.g. table row blank blank cell open parentheses 2 plus square root of 3 close parentheses open parentheses 4 plus square root of 2 close parentheses end cell end table
    table row blank equals cell 2 cross times 4 plus 2 cross times square root of 2 plus square root of 3 cross times 4 plus square root of 3 cross times square root of 2 end cell row blank equals cell 8 plus 2 square root of 2 plus 4 square root of 3 plus square root of 6 end cell end table

  • True or False?

    open parentheses a plus square root of b close parentheses squared equals a squared plus b.

    False.

    open parentheses a plus square root of b close parentheses squared not equal to a squared plus b.

    You need to expand using double brackets, open parentheses a plus square root of b close parentheses open parentheses a plus square root of b close parentheses.

    This expands and simplifies to a squared plus b plus 2 a square root of b.

  • Simplify open parentheses a plus square root of b close parentheses open parentheses a minus square root of b close parentheses.

    open parentheses a plus square root of b close parentheses open parentheses a minus square root of b close parentheses simplifies to a squared minus b.

    This is an example of the difference of two squares.

  • State what is meant by rationalising the denominator.

    Rationalising the denominator changes a fraction with surds in the denominator into an equivalent fraction where there are no surds in the denominator.

    E.g. The fraction fraction numerator 2 over denominator square root of 5 end fraction becomes fraction numerator 2 square root of 5 over denominator 5 end fraction.

  • What should you multiply the numerator and the denominator of fraction numerator 3 over denominator square root of 2 end fraction by to rationalise the denominator?

    To rationalise the denominator of fraction numerator 3 over denominator square root of 2 end fraction, you should multiply the numerator and the denominator by square root of 2.

  • What should you multiply the numerator and the denominator of fraction numerator 3 over denominator 5 minus square root of 2 end fraction by to rationalise the denominator?

    To rationalise the denominator of fraction numerator 3 over denominator 5 minus square root of 2 end fraction, you should multiply the numerator and the denominator by 5 plus square root of 2.

  • True or False?

    Multiplying the numerator and denominator of fraction numerator 2 plus square root of 3 over denominator 3 minus square root of 5 end fraction by 2 minus square root of 3 will rationalise the denominator.

    False.

    Multiplying the numerator and denominator of fraction numerator 2 plus square root of 3 over denominator 3 minus square root of 5 end fraction by 2 minus square root of 3 will not rationalise the denominator.

    You need to multiply by 3 plus square root of 5 instead.

    The multiplier should contain the same terms as the denominator but with the sign between them inverted.

  • True or False?

    Multiplying the numerator and denominator of fraction numerator 5 over denominator square root of 7 minus square root of 2 end fraction by square root of 7 minus square root of 2 will rationalise the denominator.

    False.

    Multiplying the numerator and denominator of fraction numerator 5 over denominator square root of 7 minus square root of 2 end fraction by square root of 7 minus square root of 2 will not rationalise the denominator.

    You should multiply by square root of 7 plus square root of 2 instead.

    The sign in between the two terms should be inverted.

  • To rationalise a denominator open parentheses p plus square root of q close parentheses, you multiply the numerator and the denominator by what expression?

    To rationalise a denominator open parentheses p plus square root of q close parentheses, you multiply the numerator and the denominator by open parentheses p minus square root of q close parentheses.

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    1. Topic 1

      Number Toolkit

    2. Topic 2

      Set Notation & Venn Diagrams

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      Prime Factors, HCF & LCM

    4. Topic 4

      Powers, Roots & Standard Form

    5. Topic 5

      Fractions

    6. Topic 6

      Percentages

    7. Topic 7

      Compound Interest & Depreciation

    8. Topic 8

      Fractions, Decimals & Percentages

    9. Topic 9

      Rounding, Estimation & Bounds

    10. Topic 10

      Surds

    11. Topic 11

      Using a Calculator

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      Ratio Problem Solving

    14. Topic 14

      Exchange Rates & Best Buys

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      Direct & Inverse Proportion

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