Solve the simultaneous equations. $(4^x)(8^y)=1$ $125 \div ( (\sqrt{5})^x ) = \frac{1}{\sqrt{5}}$ A cylinder has a radius of $(\sqrt{2}-1)$ cm and a volume of $\bigl(12+4\sqrt{2}\bigr)\pi\ \text{cm}^3$. Find, without using a calculator, the exact value of its height, $h$, in the form $a+b\sqrt{2}$, where $a$ and $b$ are integers.

1. Express the first equation in base 2

$(2^2)^x (2^3)^y = 1$
$2^{2x} \cdot 2^{3y} = 2^0$
$2^{2x + 3y} = 2^0$

2. Form a linear equation from the exponents of base 2

$2x + 3y = 0 \quad \text{--- (1)}$

3. Express the second equation in base 5

$5^3 \div (5^{\frac{1}{2}})^x = 5^{-\frac{1}{2}}$
$5^3 \div 5^{\frac{x}{2}} = 5^{-\frac{1}{2}}$
$5^{3 - \frac{x}{2}} = 5^{-\frac{1}{2}}$

4. Form a linear equation from the exponents of base 5 and solve for $x$

$3 - \frac{x}{2} = -\frac{1}{2}$
$\frac{x}{2} = 3 + \frac{1}{2}$
$\frac{x}{2} = \frac{7}{2}$
$x = 7$

5. Substitute $x$ into equation (1) to find $y$

Substitute $x = 7$ into (1):
$2(7) + 3y = 0$
$14 + 3y = 0$
$3y = -14$
$y = -\frac{14}{3}$

1. Set up the equation for the volume of the cylinder

The volume of a cylinder is given by $V = \pi r^2 h$.
Substituting the given values:
$\pi (\sqrt{2}-1)^2 h = (12+4\sqrt{2})\pi$
$(\sqrt{2}-1)^2 h = 12+4\sqrt{2}$

2. Expand the squared term

$(\sqrt{2}-1)^2 = (\sqrt{2})^2 - 2(\sqrt{2})(1) + 1^2$
$= 2 - 2\sqrt{2} + 1$
$= 3 - 2\sqrt{2}$
So, $(3 - 2\sqrt{2})h = 12+4\sqrt{2}$

3. Isolate $h$ and rationalise the denominator

$h = \frac{12+4\sqrt{2}}{3-2\sqrt{2}}$
Multiply the numerator and denominator by the conjugate of the denominator, $3+2\sqrt{2}$:
$h = \frac{(12+4\sqrt{2})(3+2\sqrt{2})}{(3-2\sqrt{2})(3+2\sqrt{2})}$

4. Expand and simplify to find the final answer

Denominator: $(3)^2 - (2\sqrt{2})^2 = 9 - 8 = 1$
Numerator: $12(3) + 12(2\sqrt{2}) + 4\sqrt{2}(3) + 4\sqrt{2}(2\sqrt{2})$
$= 36 + 24\sqrt{2} + 12\sqrt{2} + 8(2)$
$= 36 + 36\sqrt{2} + 16$
$= 52 + 36\sqrt{2}$
Therefore, $h = 52 + 36\sqrt{2}$