Simplify
$$\frac{5 - 2\sqrt{3}}{\sqrt{3} - 1}$$
giving your answer in the form $p + q\sqrt{3}$, where $p$ and $q$ are rational numbers. - Edexcel - A-Level Maths Pure - Question 5 - 2011 - Paper 2
Question 5
Simplify
$$\frac{5 - 2\sqrt{3}}{\sqrt{3} - 1}$$
giving your answer in the form $p + q\sqrt{3}$, where $p$ and $q$ are rational numbers.
Worked Solution & Example Answer:Simplify
$$\frac{5 - 2\sqrt{3}}{\sqrt{3} - 1}$$
giving your answer in the form $p + q\sqrt{3}$, where $p$ and $q$ are rational numbers. - Edexcel - A-Level Maths Pure - Question 5 - 2011 - Paper 2
Step 1
Multiply numerator and denominator by the conjugate
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Answer
To eliminate the square root in the denominator, we multiply the entire fraction by the conjugate of the denominator, which is 3+1:
(3−1)(3+1)(5−23)(3+1)
This simplifies the expression as follows:
Denominator:
(3)2−(1)2=3−1=2
Step 2
Expand the numerator
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Answer
Now we expand the numerator:
(5−23)(3+1)=53+5−23⋅3−23
This simplifies to:
53+5−6−23=(5−6)+(53−23)=−1+33
Step 3
Combine results
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Answer
Putting the simplified numerator and denominator together gives:
