| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2023 |
| Session | October |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indices and Surds |
| Type | Simplify algebraic expressions with indices |
| Difficulty | Easy -1.2 This is a straightforward indices manipulation question requiring only mechanical application of index laws (power rules, negative/fractional indices). Each part involves 1-2 steps of routine algebraic manipulation with no problem-solving or conceptual insight needed, making it easier than average for A-level. |
| Spec | 1.02b Surds: manipulation and rationalising denominators |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Scheme | Marks | Guidance |
| \(\frac{1}{8}x\) | B1 | \(\frac{1}{8}x\) or simplified equivalent e.g. \(0.125x\) or \(\frac{1}{8}x^1\). Accept e.g. \(\frac{x}{8}\) but \(\pm\frac{x}{8}\) is B0. Do not withhold this mark unless it is clear that they intend to write \(\frac{1}{8x}\). |
| \(\frac{1}{256}x^{\frac{3}{2}}\) | B1 | \(\frac{1}{256}x^{\frac{3}{2}}\) or simplified equivalent e.g. \(0.00390625x^{\frac{3}{2}}\). Condone \(\frac{x^{\frac{3}{2}}}{256}\). Isw after correct answer seen. Do not accept e.g. \(\frac{1}{256}\sqrt{x^3}\) or \(\frac{1}{256}(\sqrt{x})^3\) or \(\frac{1}{256}\sqrt{x}\) as \(n\) is not a simplified constant. Do not withhold this mark unless it is clear that they intend to write \(\frac{1}{256x^{\frac{3}{2}}}\). |
| \(\left(\frac{1}{2}\left\ | {\frac{1}{64}x^2 \cdot \frac{16}{\sqrt{x}}}\right\ | \right)^{\frac{4}{3}} = \left(\frac{1}{8}x^2\right)^{\frac{4}{3}} = 16x^{-2}\) |
| Answer/Scheme | Marks | Guidance |
|---|---|---|
| $\frac{1}{8}x$ | B1 | $\frac{1}{8}x$ or simplified equivalent e.g. $0.125x$ or $\frac{1}{8}x^1$. Accept e.g. $\frac{x}{8}$ but $\pm\frac{x}{8}$ is B0. Do not withhold this mark unless it is clear that they intend to write $\frac{1}{8x}$. |
| $\frac{1}{256}x^{\frac{3}{2}}$ | B1 | $\frac{1}{256}x^{\frac{3}{2}}$ or simplified equivalent e.g. $0.00390625x^{\frac{3}{2}}$. Condone $\frac{x^{\frac{3}{2}}}{256}$. Isw after correct answer seen. Do not accept e.g. $\frac{1}{256}\sqrt{x^3}$ or $\frac{1}{256}(\sqrt{x})^3$ or $\frac{1}{256}\sqrt{x}$ as $n$ is not a simplified constant. Do not withhold this mark unless it is clear that they intend to write $\frac{1}{256x^{\frac{3}{2}}}$. |
| $\left(\frac{1}{2}\left\|{\frac{1}{64}x^2 \cdot \frac{16}{\sqrt{x}}}\right\|\right)^{\frac{4}{3}} = \left(\frac{1}{8}x^2\right)^{\frac{4}{3}} = 16x^{-2}$ | M1A1 | M1: Attempts to use the index laws and proceeds to either $...x^{\frac{3}{2} \cdot \frac{4}{3}}$ or $...x^{\frac{3}{2} \cdot \frac{4}{3}}$, or $...x^{\frac{8}{3}} \div ...x^{\frac{2}{3}}$ (or $...x^{\frac{8}{3}} \div ...x^{\frac{2}{3}}$), or $...x^{-2}$, or $16x^n$ where $n \neq -\frac{1}{2}$. Allow $\frac{16}{x^2}$ to score M1. Be aware that incorrect understanding of indices such as $\frac{16}{\sqrt{x}} = 16x^{-2}$ is M0A0. A1: $16x^{-2}$ (from correct method). (Correct answer with no incorrect working seen can score M1A1). Isw after correct answer seen. $\frac{16}{x^2}$ is A0. |
**Total: 4 marks**
---\begin{enumerate}
\item Given that
\end{enumerate}
$$a = \frac { 1 } { 64 } x ^ { 2 } \quad b = \frac { 16 } { \sqrt { x } }$$
express each of the following in the form $k x ^ { n }$ where $k$ and $n$ are simplified constants.\\
(a) $a ^ { \frac { 1 } { 2 } }$\\
(b) $\frac { 16 } { b ^ { 3 } }$\\
(c) $\left( \frac { a b } { 2 } \right) ^ { - \frac { 4 } { 3 } }$
\hfill \mbox{\textit{Edexcel P1 2023 Q2 [4]}}