A-Level Edexcel Maths Pure Arithmetic Sequences & Series: (a) Write $\sqrt{45}$ in the for

(a) Write $\sqrt{45}$ in the form $\alpha\sqrt{5}$, where $\alpha$ is an integer - Edexcel - A-Level Maths Pure - Question 7 - 2006 - Paper 1

Question 7

$(a)-Write-$\sqrt{45}$-in-the-form-$\alpha\sqrt{5}$,-where-$\alpha$-is-an-integer-Edexcel-A-Level Maths Pure-Question 7-2006-Paper 1.png$

(a) Write $\sqrt{45}$ in the form $\alpha\sqrt{5}$, where $\alpha$ is an integer. (b) Express $\frac{2(3 + \sqrt{5})}{(3 - \sqrt{5})}$ in the form $b + c\sqrt{5}$, ... show full transcript

Worked Solution & Example Answer:(a) Write $\sqrt{45}$ in the form $\alpha\sqrt{5}$, where $\alpha$ is an integer - Edexcel - A-Level Maths Pure - Question 7 - 2006 - Paper 1

Step 1

Write $\sqrt{45}$ in the form $\alpha\sqrt{5}$, where $\alpha$ is an integer.

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Answer

To simplify $\sqrt{45}$ , we start by factorizing 45:

$45 = 9 \times 5 = 3^2 \times 5.$

Thus, we can express $\sqrt{45}$ as:

$\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \cdot \sqrt{5} = 3\sqrt{5}.$

Therefore, in the form $\alpha\sqrt{5}$ , we have $\alpha = 3$ .

Step 2

Express $\frac{2(3 + \sqrt{5})}{(3 - \sqrt{5})}$ in the form $b + c\sqrt{5}$, where $b$ and $c$ are integers.

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Answer

To simplify $\frac{2(3 + \sqrt{5})}{(3 - \sqrt{5})}$ , we first multiply the numerator and denominator by the conjugate of the denominator, which is $(3 + \sqrt{5})$ :

$\frac{2(3 + \sqrt{5})(3 + \sqrt{5})}{(3 - \sqrt{5})(3 + \sqrt{5})}.$

Calculating the denominator:

$(3 - \sqrt{5})(3 + \sqrt{5}) = 3^2 - (\sqrt{5})^2 = 9 - 5 = 4.$

Now for the numerator:

$2(3 + \sqrt{5})(3 + \sqrt{5}) = 2(9 + 3\sqrt{5} + 3\sqrt{5} + 5) = 2(14 + 6\sqrt{5}) = 28 + 12\sqrt{5}.$

Thus, we can rewrite the expression as:

$\frac{28 + 12\sqrt{5}}{4} = 7 + 3\sqrt{5}.$

Therefore, in the form $b + c\sqrt{5}$ , we have $b = 7$ and $c = 3$ .

(a) Write $\sqrt{45}$ in the form $\alpha\sqrt{5}$, where $\alpha$ is an integer - Edexcel - A-Level Maths Pure - Question 7 - 2006 - Paper 1

Worked Solution & Example Answer:(a) Write $\sqrt{45}$ in the form $\alpha\sqrt{5}$, where $\alpha$ is an integer - Edexcel - A-Level Maths Pure - Question 7 - 2006 - Paper 1

Write $\sqrt{45}$ in the form $\alpha\sqrt{5}$, where $\alpha$ is an integer.

Express $\frac{2(3 + \sqrt{5})}{(3 - \sqrt{5})}$ in the form $b + c\sqrt{5}$, where $b$ and $c$ are integers.

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