1. 1
   A materials scientist calculates decay factors using fractional and negative indices. She evaluates $\sqrt[3]{8^2}$, then $625^{-\frac{1}{2}}$, then $\left(\frac{1}{25}\right)^{\frac{3}{2}}$.

   (a)

   Evaluate $\sqrt[3]{8^2}$.

   [1 mark]

   • A$4$
   • B$16$
   • C$2$
   • D$64$

   (b)

   Evaluate $625^{-\frac{1}{2}}$.

   [1 mark]

   • A$25$
   • B$\frac{1}{25}$
   • C$-25$
   • D$\frac{1}{625}$

   (c)

   Evaluate $\left(\frac{1}{25}\right)^{\frac{3}{2}}$.

   [1 mark]

   • A$\frac{1}{25}$
   • B$125$
   • C$\frac{1}{125}$
   • D$\frac{3}{50}$

   Total for question 1: 3 marks

2. 2
   A designer works with surd expressions for a tiling pattern. The first task simplifies $\sqrt{8}+3\sqrt{32}$. The second task simplifies $625^{\frac{3}{4}}$ using index laws.

   (a)

   Simplify $\sqrt{8}+3\sqrt{32}$, giving your answer in the form $k\sqrt{2}$.

   [2 marks]

   (b)

   Evaluate $625^{\frac{3}{4}}$, showing your method using index laws.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A physics student rationalises denominators for exam revision. The first task is $\frac{2}{\sqrt{8}}$. The second task rationalises $\frac{1}{2-\sqrt{3}}$.

   (a)

   Rationalise the denominator of $\frac{2}{\sqrt{8}}$, giving your answer in its simplest surd form.

   [3 marks]

   (b)

   Rationalise the denominator of $\frac{1}{2-\sqrt{3}}$, giving your answer in the form $a+b\sqrt{3}$.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   An electronics engineer models signal decay using index expressions in a base of 2 and 3. The first expression is $2^{3x} \times 2^{-x+1} \div 2^{2}$. A second expression compares surd values: $\left(\sqrt{3}+1\right)^2$. A third combines both to fully rationalise and simplify a compound fraction $\frac{3}{\sqrt{3}}+\frac{1}{\sqrt{3}-1}$.

   (a)

   An electronics formula for signal decay uses $2^{3x} \times 2^{-x+1} \div 2^{2}$. Simplify this expression fully, showing every index law step, and give your answer as a single power of 2 in terms of $x$.

   [4 marks]

   (b)

   Expand and simplify $\left(\sqrt{3}+1\right)^2$ fully, giving your answer in the form $a+b\sqrt{3}$.

   [4 marks]

   (c)

   Simplify $\frac{3}{\sqrt{3}}+\frac{1}{\sqrt{3}-1}$ fully, rationalising each denominator and giving your final answer in the form $a+b\sqrt{3}$.

   [5 marks]

   Total for question 4: 13 marks

End of questions