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2. (a) Express \(\sqrt{108}\) in the form \(\alpha\sqrt{3}\), where \(a\) is an integer - Edexcel - A-Level Maths Pure - Question 4 - 2007 - Paper 2
Question 4
2. (a) Express \(\sqrt{108}\) in the form \(\alpha\sqrt{3}\), where \(a\) is an integer. (b) Express \((2 - \sqrt{3})^{2}\) in the form \(b + c\sqrt{3}\), whe... show full transcript
Worked Solution & Example Answer:2. (a) Express \(\sqrt{108}\) in the form \(\alpha\sqrt{3}\), where \(a\) is an integer - Edexcel - A-Level Maths Pure - Question 4 - 2007 - Paper 2
Step 1
Express \(\sqrt{108}\) in the form \(\alpha\sqrt{3}\), where \(a\) is an integer.
Answer
To express (\sqrt{108}) in the desired form, we start by simplifying (\sqrt{108}).
Notice that:
[
108 = 36 \times 3
]
This allows us to write:
[
\sqrt{108} = \sqrt{36 \times 3} = \sqrt{36}\sqrt{3} = 6\sqrt{3}
]
Thus, (\alpha = 6).
Step 2
Express \((2 - \sqrt{3})^{2}\) in the form \(b + c\sqrt{3}\), where \(b\) and \(c\) are integers.
Answer
To expand ((2 - \sqrt{3})^{2}), we apply the formula for the square of a binomial:
[
(a - b)^{2} = a^{2} - 2ab + b^{2}
]
Using (a = 2) and (b = \sqrt{3}), we have:
[
(2 - \sqrt{3})^{2} = 2^{2} - 2 \times 2 \times \sqrt{3} + (\sqrt{3})^{2}
]
Calculating each term:
[
= 4 - 4\sqrt{3} + 3
]
Combining the constant terms gives us:
[
= 7 - 4\sqrt{3}
]
Thus, (b = 7) and (c = -4).