### Surds Part II

Example 1

A cone has a volume of $\ (\sqrt{5} + 3) \: \text{cm}^3 $. Its base area is $\ (\sqrt{5} +1)\: \text{cm} $ What is the height of the cone ?

To find the height of the cone we divide its volume by its height

$\begin {aligned}

\frac{\sqrt{5} + 3}{\sqrt{5} +1} &= \frac{(\sqrt{5} + 3)(\sqrt{5} -1)}{(\sqrt{5} +1)(\sqrt{5} -1)}\\

&= \frac{(\sqrt{5})(\sqrt{5}) - \sqrt{5} + 3\sqrt{5} -3}{(\sqrt{5} +1)(\sqrt{5} -1)} \\

&= \frac{5+ 2\sqrt{5} - 3 }{5-1} \\

&= \frac{2+ 2\sqrt{5}}{4} \\

&= \frac{1 + \sqrt{5}}{2} \\

\end {aligned} $

Example 2

Solve the following equation:

$\ y\sqrt{24} = y\sqrt{3} + \sqrt{6} $

To solve this, we make y the subject of the equation

$\begin {aligned}

y\sqrt{24} &= y\sqrt{3} + \sqrt{6} \\

y\sqrt{24} - y\sqrt{3} &=\sqrt{6}\\

y ( \sqrt{24} - \sqrt{3} ) &= \sqrt{6}\\

y &= \frac{ \sqrt{6}}{\sqrt{24} - \sqrt{3}} \\

y &= \frac{ \sqrt{3}\sqrt{2}}{\sqrt{8}\sqrt{3} - \sqrt{3}} \\

y&= \frac{ \sqrt{2}}{\sqrt{8} - 1} \\

\end {aligned} $

Next we rationalize

$\begin {aligned}

y&= \frac{ \sqrt{2}(\sqrt{8} + 1)}{(\sqrt{8} - 1)(\sqrt{8} + 1)} \\

y&= \frac{ \sqrt{16} + \sqrt{2}}{7} \\

y&= \frac{ 4 + \sqrt{2}}{7} \\

\end {aligned} $

Example 3

Given that $\ (x + \sqrt{7})(3 + y\sqrt{7}) = 26 + 11 \sqrt{7} $, find the possible values of x and y.

To solve this question we need to expand and simplify the LHS

Then we group the terms containing surds together

$\begin {aligned}

LHS &= (x + \sqrt{7})(3 + y\sqrt{7})\\

&= 3x + xy\sqrt{7} + 3\sqrt{7} + y (\sqrt{7})(\sqrt{7}) \\

&= 3x + 7y + (3 + xy) \sqrt{7} \\

\end {aligned} $

Subsequently we equate the terms containing surds

$\ (3 + xy) \sqrt{7} = 11 \sqrt{7} $

Therefore, $\ 3 + xy = 11 $

Then we equate terms that do not contain surds

$\ 3x + 7y = 26 $

We have 2 equations that we can solve simultaneously to determine the value of x and y

$\begin {aligned}

y &= \frac{8}{x} \\

\text{Sub into second equation} \\

3x + 7(\frac{8}{x} ) &= 26 \\

3x^2 -26x + 56 &= 0 \\

(3x-14)(x-4) &= 0 \\

x = \frac{14}{3} \quad \text{or} \quad x=4 \\

\end {aligned} $

From the values of x we can then determine the value of y

$\ y = \frac{12}{7} \quad \text{or} \quad y = 2 $

Example 4

Given that $\ \sqrt{x + y \sqrt{5}} = \frac{5}{4 + \sqrt{5}} $. Determine the value of x and y.

The first step is to square both sides

$\begin {aligned}

\sqrt{x + y \sqrt{5}} &= \frac{5}{4 + \sqrt{5}} \\

\left(\sqrt{x + y \sqrt{5}}\right)^2 &= \left(\frac{5}{4 + \sqrt{5}}\right)^2 \\

x + y \sqrt{5} &= \frac{25}{(4+ \sqrt{5})^2} \\

\end {aligned} $

The second step is to simplify and rationalize the fraction on the right hand side

$\begin {aligned}

x + y \sqrt{5} &= \frac{25}{(16 + 8\sqrt{5} + 5} \\

&=\frac{25}{(21 + 8\sqrt{5}) } \\

&= \frac{25(21 + 8\sqrt{5})}{(21 + 8\sqrt{5})(21 - 8\sqrt{5}) }\\

&= \frac{525 + 200 \sqrt{5}}{441 -320} \\

&= \frac{525 + 200 \sqrt{5}}{121} \\

&= \frac{525}{121} + \frac{200}{121}\sqrt{5}\\

\end {aligned} $

Finally we compare the terms. We have a term that contains surds, $\ \frac{200}{121}\sqrt{5} $ and a term that does not contain surds $\ \frac{525}{121} $

We can equate the surd containing terms to determine the value of y

$\begin{aligned}

y\sqrt{5} &= \frac{200}{121}\sqrt{5} \\

y &= \frac{200}{121} \\

\end {aligned} $

We can also equate the terms that do not contain surds to detrmine the value of x

$\begin{aligned}

x &= \frac{525}{121} \\

\end {aligned} $

A cone has a volume of $\ (\sqrt{5} + 3) \: \text{cm}^3 $. Its base area is $\ (\sqrt{5} +1)\: \text{cm} $ What is the height of the cone ?

To find the height of the cone we divide its volume by its height

$\begin {aligned}

\frac{\sqrt{5} + 3}{\sqrt{5} +1} &= \frac{(\sqrt{5} + 3)(\sqrt{5} -1)}{(\sqrt{5} +1)(\sqrt{5} -1)}\\

&= \frac{(\sqrt{5})(\sqrt{5}) - \sqrt{5} + 3\sqrt{5} -3}{(\sqrt{5} +1)(\sqrt{5} -1)} \\

&= \frac{5+ 2\sqrt{5} - 3 }{5-1} \\

&= \frac{2+ 2\sqrt{5}}{4} \\

&= \frac{1 + \sqrt{5}}{2} \\

\end {aligned} $

Example 2

Solve the following equation:

$\ y\sqrt{24} = y\sqrt{3} + \sqrt{6} $

To solve this, we make y the subject of the equation

$\begin {aligned}

y\sqrt{24} &= y\sqrt{3} + \sqrt{6} \\

y\sqrt{24} - y\sqrt{3} &=\sqrt{6}\\

y ( \sqrt{24} - \sqrt{3} ) &= \sqrt{6}\\

y &= \frac{ \sqrt{6}}{\sqrt{24} - \sqrt{3}} \\

y &= \frac{ \sqrt{3}\sqrt{2}}{\sqrt{8}\sqrt{3} - \sqrt{3}} \\

y&= \frac{ \sqrt{2}}{\sqrt{8} - 1} \\

\end {aligned} $

Next we rationalize

$\begin {aligned}

y&= \frac{ \sqrt{2}(\sqrt{8} + 1)}{(\sqrt{8} - 1)(\sqrt{8} + 1)} \\

y&= \frac{ \sqrt{16} + \sqrt{2}}{7} \\

y&= \frac{ 4 + \sqrt{2}}{7} \\

\end {aligned} $

Example 3

Given that $\ (x + \sqrt{7})(3 + y\sqrt{7}) = 26 + 11 \sqrt{7} $, find the possible values of x and y.

To solve this question we need to expand and simplify the LHS

Then we group the terms containing surds together

$\begin {aligned}

LHS &= (x + \sqrt{7})(3 + y\sqrt{7})\\

&= 3x + xy\sqrt{7} + 3\sqrt{7} + y (\sqrt{7})(\sqrt{7}) \\

&= 3x + 7y + (3 + xy) \sqrt{7} \\

\end {aligned} $

Subsequently we equate the terms containing surds

$\ (3 + xy) \sqrt{7} = 11 \sqrt{7} $

Therefore, $\ 3 + xy = 11 $

Then we equate terms that do not contain surds

$\ 3x + 7y = 26 $

We have 2 equations that we can solve simultaneously to determine the value of x and y

$\begin {aligned}

y &= \frac{8}{x} \\

\text{Sub into second equation} \\

3x + 7(\frac{8}{x} ) &= 26 \\

3x^2 -26x + 56 &= 0 \\

(3x-14)(x-4) &= 0 \\

x = \frac{14}{3} \quad \text{or} \quad x=4 \\

\end {aligned} $

From the values of x we can then determine the value of y

$\ y = \frac{12}{7} \quad \text{or} \quad y = 2 $

Example 4

Given that $\ \sqrt{x + y \sqrt{5}} = \frac{5}{4 + \sqrt{5}} $. Determine the value of x and y.

The first step is to square both sides

$\begin {aligned}

\sqrt{x + y \sqrt{5}} &= \frac{5}{4 + \sqrt{5}} \\

\left(\sqrt{x + y \sqrt{5}}\right)^2 &= \left(\frac{5}{4 + \sqrt{5}}\right)^2 \\

x + y \sqrt{5} &= \frac{25}{(4+ \sqrt{5})^2} \\

\end {aligned} $

The second step is to simplify and rationalize the fraction on the right hand side

$\begin {aligned}

x + y \sqrt{5} &= \frac{25}{(16 + 8\sqrt{5} + 5} \\

&=\frac{25}{(21 + 8\sqrt{5}) } \\

&= \frac{25(21 + 8\sqrt{5})}{(21 + 8\sqrt{5})(21 - 8\sqrt{5}) }\\

&= \frac{525 + 200 \sqrt{5}}{441 -320} \\

&= \frac{525 + 200 \sqrt{5}}{121} \\

&= \frac{525}{121} + \frac{200}{121}\sqrt{5}\\

\end {aligned} $

Finally we compare the terms. We have a term that contains surds, $\ \frac{200}{121}\sqrt{5} $ and a term that does not contain surds $\ \frac{525}{121} $

We can equate the surd containing terms to determine the value of y

$\begin{aligned}

y\sqrt{5} &= \frac{200}{121}\sqrt{5} \\

y &= \frac{200}{121} \\

\end {aligned} $

We can also equate the terms that do not contain surds to detrmine the value of x

$\begin{aligned}

x &= \frac{525}{121} \\

\end {aligned} $

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