A-level Mathematics/OCR/C1/Indices and Surds/Solutions

These are the solutions to Indices and Surd Problems.

Indices

1. ${\displaystyle x^{5}}$
2. ${\displaystyle x}$
3. ${\displaystyle x^{3}}$
4. ${\displaystyle {\frac {1}{x^{4}}}}$
5. ${\displaystyle x^{12}}$
6. ${\displaystyle x^{2}y^{2}}$
7. ${\displaystyle {\frac {y^{3}}{b^{3}}}}$
8. ${\displaystyle {\sqrt[{3}]{x^{2}}}}$
9. ${\displaystyle x^{2}}$
10. ${\displaystyle 1}$

Surds

1. ${\displaystyle 3}$
2. ${\displaystyle 5}$
3. ${\displaystyle 3{\sqrt {5}}}$
4. ${\displaystyle 3{\sqrt {7}}}$
5. ${\displaystyle 10}$
6. ${\displaystyle 4{\sqrt {3}}}$
7. ${\displaystyle 7{\sqrt {2}}}$
8. ${\displaystyle 5{\sqrt {3}}}$
9. ${\displaystyle 4{\sqrt {2}}}$
10. ${\displaystyle 8{\sqrt {5}}}$
11. ${\displaystyle 16}$
12. ${\displaystyle {\frac {1}{2}}}$
13. ${\displaystyle {\sqrt[{3}]{\frac {27}{125}}}}$ = ${\displaystyle {\frac {27^{\frac {1}{3}}}{125^{\frac {1}{3}}}}}$ = ${\displaystyle {\frac {3}{5}}}$ (Note: ${\displaystyle x^{\frac {1}{3}}\times x^{\frac {1}{3}}\times x^{\frac {1}{3}}=x}$)
14. ${\displaystyle {\sqrt {\frac {162}{200}}}}$ = ${\displaystyle {\frac {\sqrt {162}}{\sqrt {200}}}}$ = ${\displaystyle {\frac {{\sqrt {81}}\times {\sqrt {2}}}{{\sqrt {100}}\times {\sqrt {2}}}}}$ = ${\displaystyle {\frac {9{\sqrt {2}}}{10{\sqrt {2}}}}}$ = ${\displaystyle {\frac {9}{10}}}$
15. ${\displaystyle 5}$
16. ${\displaystyle {\frac {\sqrt {14}}{3}}}$
17. ${\displaystyle 1}$ (denominator is already rational)
18. ${\displaystyle -{\frac {32-8{\sqrt {26}}}{10}}}$
19. ${\displaystyle {\frac {144+24{\sqrt {19}}}{68}}}$
20. ${\displaystyle {\frac {9+3{\sqrt {65}}}{-56}}}$
21. ${\displaystyle {\frac {3}{4}}}$
22. ${\displaystyle {\frac {{a^{2}}-a{\sqrt {a}}}{a-1}}}$