Surds-Rationalising the denominator (worksheets with solutions ... - Free Printable

Let's solve each of these problems step by step. These are all rationalizing the denominator problems, where we eliminate radicals from the denominator using conjugates or simplification.

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Problem 9) $\frac{4}{9\sqrt{2}}$

We want to rationalize the denominator. The denominator is $9\sqrt{2}$, so we multiply numerator and denominator by $\sqrt{2}$:

$$
\frac{4}{9\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{4\sqrt{2}}{9 \cdot 2} = \frac{4\sqrt{2}}{18}
$$

Simplify the fraction:

$$
\frac{4\sqrt{2}}{18} = \frac{2\sqrt{2}}{9}
$$

✔ Answer: $\boxed{\frac{2\sqrt{2}}{9}}$

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Problem 10) $\frac{5}{1 + \sqrt{2}}$

Here, the denominator has a binomial with a radical. We use the conjugate: $1 - \sqrt{2}$

Multiply numerator and denominator by the conjugate:

$$
\frac{5}{1 + \sqrt{2}} \cdot \frac{1 - \sqrt{2}}{1 - \sqrt{2}} = \frac{5(1 - \sqrt{2})}{(1 + \sqrt{2})(1 - \sqrt{2})}
$$

Denominator: difference of squares:

$$
(1)^2 - (\sqrt{2})^2 = 1 - 2 = -1
$$

Numerator: $5(1 - \sqrt{2}) = 5 - 5\sqrt{2}$

So:

$$
\frac{5 - 5\sqrt{2}}{-1} = -5 + 5\sqrt{2} = 5\sqrt{2} - 5
$$

✔ Answer: $\boxed{5\sqrt{2} - 5}$

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Problem 11) $\frac{6}{\sqrt{3} + 4}$

Again, denominator is a binomial with a radical. Use conjugate: $\sqrt{3} - 4$

Multiply numerator and denominator by $\sqrt{3} - 4$:

$$
\frac{6}{\sqrt{3} + 4} \cdot \frac{\sqrt{3} - 4}{\sqrt{3} - 4} = \frac{6(\sqrt{3} - 4)}{(\sqrt{3} + 4)(\sqrt{3} - 4)}
$$

Denominator: $(\sqrt{3})^2 - (4)^2 = 3 - 16 = -13$

Numerator: $6\sqrt{3} - 24$

So:

$$
\frac{6\sqrt{3} - 24}{-13} = -\frac{6\sqrt{3}}{13} + \frac{24}{13} = \frac{24 - 6\sqrt{3}}{13}
$$

✔ Answer: $\boxed{\frac{24 - 6\sqrt{3}}{13}}$

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Problem 12) $\frac{7}{2 - \sqrt{5}}$

Use conjugate: $2 + \sqrt{5}$

Multiply numerator and denominator:

$$
\frac{7}{2 - \sqrt{5}} \cdot \frac{2 + \sqrt{5}}{2 + \sqrt{5}} = \frac{7(2 + \sqrt{5})}{(2 - \sqrt{5})(2 + \sqrt{5})}
$$

Denominator: $2^2 - (\sqrt{5})^2 = 4 - 5 = -1$

Numerator: $14 + 7\sqrt{5}$

So:

$$
\frac{14 + 7\sqrt{5}}{-1} = -14 - 7\sqrt{5}
$$

✔ Answer: $\boxed{-14 - 7\sqrt{5}}$

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✔ Final Answers:

9) $\boxed{\frac{2\sqrt{2}}{9}}$
10) $\boxed{5\sqrt{2} - 5}$
11) $\boxed{\frac{24 - 6\sqrt{3}}{13}}$
12) $\boxed{-14 - 7\sqrt{5}}$

Let me know if you'd like explanations in a different format!