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

These are the solutions to Indices and Surd Problems.

Indices

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

Surds

1. ${\sqrt {9}}=3$
2. ${\sqrt {25}}=5$
3. ${\sqrt {45}}={\sqrt {9}}{\sqrt {5}}=3{\sqrt {5}}$
4. $3{\sqrt {7}}$
5. ${\sqrt {100}}=10$
6. $4{\sqrt {3}}$
7. $7{\sqrt {2}}$
8. $5{\sqrt {3}}$
9. $4{\sqrt {2}}$
10. $8{\sqrt {5}}$
11. $16$
12. ${\frac {1}{2}}$
13. ${\sqrt[{3}]{\frac {27}{125}}}$ = ${\frac {27^{\frac {1}{3}}}{125^{\frac {1}{3}}}}$ = ${\frac {3}{5}}$ (Note: $x^{\frac {1}{3}}\times x^{\frac {1}{3}}\times x^{\frac {1}{3}}=x$)
14. ${\sqrt {\frac {162}{200}}}$ = ${\frac {\sqrt {162}}{\sqrt {200}}}$ = ${\frac {{\sqrt {81}}\times {\sqrt {2}}}{{\sqrt {100}}\times {\sqrt {2}}}}$ = ${\frac {9{\sqrt {2}}}{10{\sqrt {2}}}}$ = ${\frac {9}{10}}$
15. $5$
16. ${\frac {\sqrt {14}}{3}}$
17. $1$ (denominator is already rational)
18. $-{\frac {32-8{\sqrt {26}}}{10}}$
19. ${\frac {144+24{\sqrt {19}}}{68}}$
20. ${\frac {9+3{\sqrt {65}}}{-56}}$
21. ${\frac {3}{4}}$
22. ${\frac {{a^{2}}-a{\sqrt {a}}}{a-1}}$