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3. (i) Express \( (5 - \sqrt{8})(1 + \sqrt{2}) \) in the form \( a + b\sqrt{2} \), where \( a \) and \( b \) are integers - Edexcel - A-Level Maths Pure - Question 5 - 2013 - Paper 3
Question 5
3. (i) Express \( (5 - \sqrt{8})(1 + \sqrt{2}) \) in the form \( a + b\sqrt{2} \), where \( a \) and \( b \) are integers. (ii) Express \( \sqrt{80} + \dfrac{30}{\... show full transcript
Worked Solution & Example Answer:3. (i) Express \( (5 - \sqrt{8})(1 + \sqrt{2}) \) in the form \( a + b\sqrt{2} \), where \( a \) and \( b \) are integers - Edexcel - A-Level Maths Pure - Question 5 - 2013 - Paper 3
Step 1
Express \( (5 - \sqrt{8})(1 + \sqrt{2}) \) in the form \( a + b\sqrt{2} \)
Answer
To solve this, we will expand the expression:
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Apply the distributive property: [(5 - \sqrt{8})(1 + \sqrt{2}) = 5 \cdot 1 + 5 \cdot \sqrt{2} - \sqrt{8} \cdot 1 - \sqrt{8} \cdot \sqrt{2}] [= 5 + 5\sqrt{2} - \sqrt{8} - \sqrt{16}] [= 5 + 5\sqrt{2} - 2\sqrt{2} - 4]
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Combine like terms: [= (5 - 4) + (5 - 2)\sqrt{2} = 1 + 3\sqrt{2}]
Thus, we have ( a = 1 ) and ( b = 3 ).
Step 2
Express \( \sqrt{80} + \dfrac{30}{\sqrt{5}} \) in the form \( c\sqrt{5} \)
Answer
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Simplify ( \sqrt{80} ): [\sqrt{80} = \sqrt{16 \cdot 5} = \sqrt{16} \cdot \sqrt{5} = 4\sqrt{5}]
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Rationalize the denominator for ( \dfrac{30}{\sqrt{5}} ): [\dfrac{30}{\sqrt{5}} \cdot \dfrac{\sqrt{5}}{\sqrt{5}} = \dfrac{30\sqrt{5}}{5} = 6\sqrt{5}]
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Combine like terms: [4\sqrt{5} + 6\sqrt{5} = (4 + 6)\sqrt{5} = 10\sqrt{5}]
Thus, we have ( c = 10 ).