Q1
Answer
Write the following in the form $a\sqrt{b}$, where $b$ is a prime number:
  1. $\sqrt{300}-\sqrt{48}$
  2. $\sqrt{37.5}$
  1. $6\sqrt{3}$
  2. $\frac{5}{2}\sqrt{6}$
Q2
Answer
Write the following in the form $a+b\sqrt{c}$:
  1. $(7+\sqrt{5})(3-\sqrt{5})$
  2. $\dfrac{\sqrt{2}}{1-\sqrt{2}}$
  1. $16-4\sqrt{5}$
  2. $-\sqrt{2}-2$
Q3
Answer
Simplify
  1. $\dfrac{\sqrt{48}-\sqrt{600}}{\sqrt{12}}$
  2. $\dfrac{\sqrt{2}}{4+3\sqrt{2}}$
  1. $2-5\sqrt{2}$
  2. $3-2\sqrt{2}$
Q4
Answer
Simplify
  1. $\sqrt{48}-\dfrac{6}{\sqrt{3}}$
  2. $\dfrac{12\sqrt{3}}{\sqrt{50}-\sqrt{18}}$
  1. $2\sqrt{3}$
  2. $3\sqrt{6}$
Q5
Answer
Simplify
  1. $\left(\dfrac{25x^4}{4}\right)^{-\frac{1}{2}}$
  2. $\dfrac{(32x^3)^{\frac{2}{5}}}{2x\sqrt{x}}$
  1. $\dfrac{2}{5x^2}$
  2. $2x^{-\frac{3}{10}}$
Q6
Answer
Solve $32\sqrt{2}=2^x$.
$x = \dfrac{11}{2}$
Q7
Answer
Given that $(2+\sqrt{5})x=6-\sqrt{5}$, find $x$ in the form $a+b\sqrt{5}$.
$8\sqrt{5}-17$
Q8
Answer
Solve:
  1. $10+x\sqrt{8}=\dfrac{6x}{\sqrt{2}}$
  2. $5^x\times5^{x+4}=25$
  1. $x = 5\sqrt{2}$
  2. $x = -1$
Q9
Answer
Given that $y = \dfrac{1}{64}x^3$, find $4y^{-1}$ in the form $kx^n$.
$256x^{-3}$
Q10
Answer
Write:
  1. $2^6\times2^2$ in the form $8^n$
  2. $5\times4^{\frac{2}{3}}+3\times16^{\frac{1}{3}}$ in the form $2^n$
  1. $8^{\frac{8}{3}}$
  2. $2^{\frac{13}{3}}$
Q11
Answer
Find the integer $n$ such that $$n<5\sqrt{3}<n+1$$
$n=8$
Q12
Answer
The area of a rectangle of width $3\sqrt{2}-3$ is 6. Find the height of the rectangle.
$2\sqrt{2}+2$
Q13
Answer
Show that $\left(\dfrac{4}{3}\right)^{\frac{1}{2}}+\left(\dfrac{1}{3}\right)^{-\frac{1}{2}}$ can be written in the form $\dfrac{a}{b}\sqrt{c}$.
$\dfrac{5}{3}\sqrt{3}$
Q14
Answer
Given that $y=2^x$, express the following in terms of $y$:
  1. $2^{2x-1}$
  2. $8^x-4^{-x}$
  1. $\dfrac{y^2}{2}$
  2. $y^3 - \dfrac{1}{y^2}$
Q15
Answer
Write the following in the form $2^n$:
  1. $\Big(\big(2^3\big)^2\Big)^3$
  2. $\big(2^3\big)^{\big(2^3\big)}$
  3. $2^{\Big(\big(3^2\big)^3\Big)}$
  4. $2^{\Big(3^{\big(2^3\big)}\Big)}$
  1. 18
  2. 24
  3. 729
  4. 6561
Q16
Answer
A triangle $ABC$ has $AB=BC=4+\sqrt{3}$ and $AC=4+4\sqrt{3}$. $M$ is the midpoint of $AC$. Find the exact length of $BM$ and hence find the area of the triangle.
$BM=\sqrt{3}$, area $=6+2\sqrt{3}$
Q17
Answer
By first expanding $(1-\sqrt{5})^2$, solve $y^2=3-\sqrt{5}$.
$y = \pm \dfrac{\sqrt{2}-\sqrt{10}}{2}$
Q18
Answer
A triangle $ABC$ has $AB = 2\sqrt{3}-1$, $BC = \sqrt{3}+2$, and $\angle ABC = 90^{\circ}$.
  1. Find the area of the triangle.
  2. Show that $AC = 2\sqrt{5}$.
  3. Find the exact value of $\tan(\angle ACB)$.
  1. $2+\dfrac{3\sqrt{3}}{2}$
  2. Algebra - see video
  3. $\tan(\angle ACB)=5\sqrt{3}-8$
Q19
Answer
Solve $$25^{2x-1}=0.2^{x+3}$$
$x = -\frac{1}{5}$
Q20
Answer
Simplify $$\sqrt{\dfrac{3+2\sqrt{2}}{3-2\sqrt{2}}}$$
$3+2\sqrt{2}$
Q21
Answer
Evaluate $$\dfrac{1}{\sqrt{1}+\sqrt{2}} + \dfrac{1}{\sqrt{2}+\sqrt{3}} + ... + \dfrac{1}{\sqrt{24}+\sqrt{25}}$$
$4$
Q22
Answer
Given that $(a+b\sqrt{2})^2=59+30\sqrt{2}$, and that $a$ and $b$ are both positive, find $a$ and $b$.
$a = 3$ and $b = 5$
Q23
Answer
  1. Express $\sqrt[3]{24}$ in the form $a\sqrt[3]{3}$.
  2. Hence find the integer $n$ such that $$\sqrt[3]{n} = \sqrt[3]{24}+\sqrt[3]{81}$$
  1. $\sqrt[3]{8}\times\sqrt[3]{3} = 2\sqrt[3]{3}$
  2. $375$
Q24
Answer
By considering $(a+b\sqrt{2})^2$, find $\sqrt{17-12\sqrt{2}}$.
$3-2\sqrt{2}$