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Rationalise the denominator of the following: 3.1.1 4/3√5 3.1.2 2/√6-2

Mathematics

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Answer

4515\mathbf{\frac{4\sqrt{5}}{15}}

6+2\mathbf{\sqrt{6}+2}

Step-by-step solution

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let's tackle Question 3.

QUESTION 3

3.1.1 Rationalise the denominator of 435\frac{4}{3\sqrt{5}}.

  1. Multiply the numerator and denominator by 5\sqrt{5} to eliminate the radical in the denominator.

    435×55\frac{4}{3\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}}

  2. Perform the multiplication.

    453×(5×5)=453×5=4515\frac{4\sqrt{5}}{3 \times (\sqrt{5} \times \sqrt{5})} = \frac{4\sqrt{5}}{3 \times 5} = \frac{4\sqrt{5}}{15}

    The rationalized expression is 4515\boxed{\mathbf{\frac{4\sqrt{5}}{15}}}.

    3.1.2 Rationalise the denominator of 26−2\frac{2}{\sqrt{6}-2}.

  3. Multiply the numerator and denominator by the conjugate of the denominator, which is 6+2\sqrt{6}+2.

    26−2×6+26+2\frac{2}{\sqrt{6}-2} \times \frac{\sqrt{6}+2}{\sqrt{6}+2}

  4. Multiply the numerators and denominators.

    Remember that (a−b)(a+b)=(a)2−b2(\sqrt{a}-b)(\sqrt{a}+b) = (\sqrt{a})^2 - b^2.

    2(6+2)(6)2−(2)2\frac{2(\sqrt{6}+2)}{(\sqrt{6})^2 - (2)^2}

  5. Simplify the expression.

    26+2×26−4=26+42\frac{2\sqrt{6} + 2 \times 2}{6 - 4} = \frac{2\sqrt{6} + 4}{2}

  6. Divide both terms in the numerator by the denominator.

    262+42=6+2\frac{2\sqrt{6}}{2} + \frac{4}{2} = \sqrt{6} + 2

    The rationalized expression is 6+2\boxed{\mathbf{\sqrt{6}+2}}.

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