Rationalising the Denominator
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Questions and Answers

What is the result of multiplying the numerator and denominator of the fraction 1/(√2 + 1) by the conjugate of √2 + 1?

  • (√2 - 1)/(√2 + 1)^2
  • (√2 - 1)/(√2 + 1)(√2 - 1) (correct)
  • (√2 + 1)/(√2 - 1)
  • (√2 + 1)/(√2 + 1)(√2 - 1)

    • What is the purpose of rationalising the denominator in a fraction?

  • To eliminate the surds from the numerator
  • To make the fraction more complex and challenging
  • To eliminate the surds from the entire fraction
  • To simplify the fraction and make it easier to work with (correct)

    • What is the simplified form of the surd √(18)?

  • √2√3
  • 3√2
  • 3√3 (correct)
  • 2√3

    • What is the result of adding 2√5 and 3√5?

    <p>5√5</p> Signup and view all the answers

    What is the result of multiplying ∛2 and ∛4?

    <p>∛8</p> Signup and view all the answers

    What is the goal of rationalising the denominator in a fraction?

    <p>To get a rational denominator</p> Signup and view all the answers

    Surds can be added or subtracted only if they have the same value.

    <p>False</p> Signup and view all the answers

    What is the rule for simplifying surds?

    <p>Removing any perfect squares from the surd and combining like terms.</p> Signup and view all the answers

    When multiplying surds with different orders, the resulting surd will have the ______________ order.

    <p>highest</p> Signup and view all the answers

    Match the following operations with the correct rule:

    <p>Adding surds = Can be added or subtracted only if they have the same order Multiplying surds = Multiply the numbers inside the surd and apply the rules of indices Rationalising the denominator = Multiply the numerator and denominator by the conjugate of the denominator Simplifying surds = Removing any perfect squares from the surd and combining like terms</p> Signup and view all the answers

    Study Notes

    Rationalising The Denominator

    • Rationalising the denominator means eliminating the surds (radicals) from the denominator of a fraction.
    • This is done to simplify the fraction and make it easier to work with.
    • The process involves multiplying the numerator and denominator by the conjugate of the denominator.
    • The conjugate of a surd is a value that, when multiplied by the surd, eliminates the surd.

    Example: Rationalise the denominator of 1/(√2 + 1):

    • Multiply numerator and denominator by √2 - 1 (conjugate of √2 + 1)
    • 1/(√2 + 1) = (√2 - 1)/(√2 + 1)(√2 - 1) = (√2 - 1)/((√2)^2 - 1^2) = (√2 - 1)/1

    Simplifying Surds

    • Simplifying surds involves expressing them in their simplest form.
    • This is done by removing any perfect squares from the surd.
    • A perfect square is a value that can be expressed as n^2, where n is an integer.

    Example: Simplify √8:

    • √8 = √(4 × 2) = √4 × √2 = 2√2

    Addition And Subtraction Of Surds

    • Surds can be added and subtracted only if they have the same value under the surd.
    • When adding or subtracting surds, combine like terms.

    Example: Simplify 2√3 + 3√3:

    • 2√3 + 3√3 = (2 + 3)√3 = 5√3

    Multiplication Of Surds

    • Surds can be multiplied by multiplying the values under the surd and adding the indices.
    • When multiplying surds, the order of the surds does not matter.

    Example: Simplify √2 × √3:

    • √2 × √3 = √(2 × 3) = √6

    Rationalising Cube Roots

    • Rationalising cube roots involves eliminating the cube roots from the denominator of a fraction.
    • This is done by multiplying the numerator and denominator by the conjugate of the denominator.
    • The conjugate of a cube root is a value that, when multiplied by the cube root, eliminates the cube root.

    Example: Rationalise the denominator of 1/(∛2 + 1):

    • Multiply numerator and denominator by ∛4 - 1 (conjugate of ∛2 + 1)
    • 1/(∛2 + 1) = (∛4 - 1)/(∛2 + 1)(∛4 - 1) = (∛4 - 1)/((∛2)^3 + 1^3) = (∛4 - 1)/3

    Rationalising the Denominator

    • Rationalising the denominator is the process of removing surds from the denominator of a fraction to simplify it and make it easier to work with.
    • The goal is to get a rational denominator, i.e., a denominator without surds.

    Steps to Rationalise the Denominator

    • Multiply the numerator and denominator by the conjugate of the denominator.
    • Simplify the resulting fraction.

    Simplifying Surds

    • Surds are irrational numbers that cannot be expressed as a finite decimal or fraction.
    • Simplifying surds involves expressing them in their simplest form by: • Removing any perfect squares from the surd. • Combining like terms.

    Operations with Surds

    • Surds can be added or subtracted only if they have the same order (i.e., same radical index).
    • The rules for adding and subtracting surds are similar to those for adding and subtracting algebraic terms.
    • Example: √2 + √2 = 2√2, but √2 + √3 cannot be simplified further.

    Multiplication of Surds

    • Surds can be multiplied by multiplying the numbers inside the surd and applying the rules of indices.
    • Example: √2 × √3 = √(2 × 3) = √6
    • When multiplying surds with different orders, the resulting surd will have the highest order.

    Rationalising Cube Roots

    • Rationalising cube roots involves removing cube roots from the denominator of a fraction.
    • This can be done by multiplying the numerator and denominator by the conjugate of the denominator, which is the cube of the denominator.
    • Example: 1 / (√2) = (√2^2) / (√2^3) = (√4) / 2

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    Description

    Learn how to simplify fractions by eliminating surds from the denominator using conjugates. Understand the process and practice rationalising the denominator with examples.

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