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(a) Write $ oot{5}{4}$ in the form $ ext{a} oot{5}{5}$, where $ ext{a}$ is an integer - Edexcel - A-Level Maths Pure - Question 8 - 2006 - Paper 1
Question 8
(a) Write $ oot{5}{4}$ in the form $ ext{a} oot{5}{5}$, where $ ext{a}$ is an integer. (b) Express $ rac{2 ext{a}+ oot{5}{5}}{(3- oot{5}{5})}$ in the form $ ext{b}+... show full transcript
Worked Solution & Example Answer:(a) Write $ oot{5}{4}$ in the form $ ext{a} oot{5}{5}$, where $ ext{a}$ is an integer - Edexcel - A-Level Maths Pure - Question 8 - 2006 - Paper 1
Step 1
Write $ oot{5}{4}$ in the form $ ext{a} oot{5}{5}$, where $ ext{a}$ is an integer.
Answer
To express in the required form, we can first simplify it. We know:
oot{5}{4} = oot{5}{4 imes 5} / oot{5}{5} = rac{ oot{5}{20}}{ oot{5}{5}}$$ Thus, we see that if $ ext{a} = 3$, then:oot{5}{4} = 3 oot{5}{5}$$
Step 2
Express $ rac{2 ext{a}+ oot{5}{5}}{(3- oot{5}{5})}$ in the form $ ext{b}+ oot{5}{c}$, where $ ext{b}$ and $ ext{c}$ are integers.
Answer
To begin with, we need to rationalize the denominator of the expression:
oot{5}{5}}{(3- oot{5}{5})} imes rac{(3+ oot{5}{5})}{(3+ oot{5}{5})$$ This simplifies to: $$ rac{(2 ext{a}+ oot{5}{5})(3+ oot{5}{5})}{3^2 - ( oot{5}{5})^2} = rac{(2 ext{a}+ oot{5}{5})(3+ oot{5}{5})}{9 - 5} = rac{(2 ext{a}+ oot{5}{5})(3+ oot{5}{5})}{4}$$ Next, let's expand the numerator: $$= rac{(6 ext{a} + 2 ext{a} oot{5}{5} + 3 oot{5}{5} + 5)}{4}$$ Now separate the terms: $$= rac{6 ext{a} + 5}{4} + rac{(2 ext{a} + 3) oot{5}{5}}{4}$$ Identifying $ ext{b}$ and $ ext{c}$ respectively, we find: - $ ext{b} = rac{6 ext{a} + 5}{4}$ - $ ext{c} = rac{2 ext{a} + 3}{4}$