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(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 6 - 2013 - Paper 3
Question 6
(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} + \frac{30}{\sqrt... show full transcript
Worked Solution & Example Answer:(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 6 - 2013 - Paper 3
Step 1
Express \( (5 - \sqrt{8})(1 + \sqrt{2}) \) in the form \( a + b\sqrt{2} \)
Answer
To solve ( (5 - \sqrt{8})(1 + \sqrt{2}) ), we first expand the expression:
[ (5 - \sqrt{8})(1 + \sqrt{2}) = 5 \cdot 1 + 5 \cdot \sqrt{2} - \sqrt{8} \cdot 1 - \sqrt{8} \cdot \sqrt{2} ]
This simplifies to:
[ 5 + 5\sqrt{2} - \sqrt{8} - \sqrt{8 \cdot 2} ]
Now substituting ( \sqrt{8} = 2\sqrt{2} ):
[ 5 + 5\sqrt{2} - 2\sqrt{2} - \sqrt{16} = 5 + (5 - 2)\sqrt{2} - 4 ]
Combining like terms:
[ (5 - 4) + 3\sqrt{2} = 1 + 3\sqrt{2} ]
Thus, ( a = 1 ) and ( b = 3 ).
Step 2
Express \( \sqrt{80} + \frac{30}{\sqrt{5}} \) in the form \( c\sqrt{5} \)
Answer
To express ( \sqrt{80} + \frac{30}{\sqrt{5}} ):
First, simplify ( \sqrt{80} ):
[ \sqrt{80} = \sqrt{16 \cdot 5} = 4\sqrt{5} ]
Now, for the term ( \frac{30}{\sqrt{5}} ), we can rationalize the denominator:
[ \frac{30}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{30\sqrt{5}}{5} = 6\sqrt{5} ]
Adding these two terms together gives:
[ 4\sqrt{5} + 6\sqrt{5} = (4 + 6)\sqrt{5} = 10\sqrt{5} ]
Thus, ( c = 10 ).