Express the following surd fractions with rational denominators:

Question

1. Root 7 minus 2, all on root 7 plus 2.
2. 3 root 6 minus 2, all on 3 root 6 plus 2.

Answer 1

To express the given surd fractions with rational denominators, we'll multiply both the numerator and denominator by the conjugate of the denominator.

1. \(\frac{\sqrt{7} - 2}{\sqrt{7} + 2}\)

The conjugate of the denominator \(\sqrt{7} + 2\) is \(\sqrt{7} - 2\).

Now we can multiply the numerator and denominator:

\[ \frac{\sqrt{7} - 2}{\sqrt{7} + 2} \cdot \frac{\sqrt{7} - 2}{\sqrt{7} - 2} = \frac{(\sqrt{7} - 2)^2}{(\sqrt{7} + 2)(\sqrt{7} - 2)} \]

Calculating the denominator: \[ (\sqrt{7} + 2)(\sqrt{7} - 2) = 7 - 4 = 3 \]

Now for the numerator: \[ (\sqrt{7} - 2)^2 = 7 - 4\sqrt{7} + 4 = 11 - 4\sqrt{7} \]

Putting it all together: \[ \frac{11 - 4\sqrt{7}}{3} \]

2. \(\frac{3\sqrt{6} - 2}{3\sqrt{6} + 2}\)

The conjugate of the denominator \(3\sqrt{6} + 2\) is \(3\sqrt{6} - 2\).

Now we can multiply the numerator and denominator:

\[ \frac{3\sqrt{6} - 2}{3\sqrt{6} + 2} \cdot \frac{3\sqrt{6} - 2}{3\sqrt{6} - 2} = \frac{(3\sqrt{6} - 2)^2}{(3\sqrt{6} + 2)(3\sqrt{6} - 2)} \]

Calculating the denominator: \[ (3\sqrt{6} + 2)(3\sqrt{6} - 2) = (3\sqrt{6})^2 - 2^2 = 54 - 4 = 50 \]

Now for the numerator: \[ (3\sqrt{6} - 2)^2 = (3\sqrt{6})^2 - 2 \cdot 3\sqrt{6} \cdot 2 + 2^2 = 54 - 12\sqrt{6} + 4 = 58 - 12\sqrt{6} \]

Putting it all together: \[ \frac{58 - 12\sqrt{6}}{50} \]

Final Answers:

1. \(\frac{11 - 4\sqrt{7}}{3}\)
2. \(\frac{58 - 12\sqrt{6}}{50}\)

Answer 2

How did you calculate the denominator for the first question.

Answer 3

Let's carefully break down the calculation of the denominator for the first surd fraction, which is:

\[ \frac{\sqrt{7} - 2}{\sqrt{7} + 2} \]

To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator:

\[ \sqrt{7} + 2 \Rightarrow \text{conjugate: } \sqrt{7} - 2 \]

So we have:

\[ \frac{\sqrt{7} - 2}{\sqrt{7} + 2} \cdot \frac{\sqrt{7} - 2}{\sqrt{7} - 2} \]

The denominator in this expression becomes:

\[ (\sqrt{7} + 2)(\sqrt{7} - 2) \]

This is a difference of squares formula, \(a^2 - b^2\), where:

• \(a = \sqrt{7}\)
• \(b = 2\)

Calculating it:

\[ (\sqrt{7})^2 - (2)^2 = 7 - 4 = 3 \]

So the denominator simplifies correctly to:

\[ 3 \]

Thus, the correct calculation for the denominator is:

\[ \sqrt{7}^2 - 2^2 = 7 - 4 = 3 \]

The result matches what we found earlier. Thank you for your question, and I'm glad we could clarify this detail!

Answer 4

Thanks that was very helpful. 🫶🏼

Answer 5

You're very welcome! I'm glad I could help. If you have any more questions or need further assistance, feel free to ask. Happy studying! 🫶🏼