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

Feel free to add more problems as long as you provide the solutions

Indices

Simplify the Indices.

1. ${\displaystyle x^{2}x^{3}}$
2. ${\displaystyle x^{3}x^{-2}}$
3. ${\displaystyle {\frac {x^{5}}{x^{2}}}}$
4. ${\displaystyle x^{-4}}$
5. ${\displaystyle \left(x^{3}\right)^{4}}$
6. ${\displaystyle \left(xy\right)^{2}}$
7. ${\displaystyle \left({\frac {y}{b}}\right)^{3}}$
8. ${\displaystyle x^{\frac {2}{3}}}$
9. ${\displaystyle {\frac {x^{2}x^{4}}{\left(x^{2}\right)^{2}}}}$
10. ${\displaystyle x^{5}x^{-5}}$

Surds

Simplify the following Surds.

1. ${\displaystyle {\sqrt {9}}}$
2. ${\displaystyle {\sqrt {25}}}$
3. ${\displaystyle {\sqrt {45}}}$
4. ${\displaystyle {\sqrt {63}}}$
5. ${\displaystyle {\sqrt {100}}}$
6. ${\displaystyle {\sqrt {48}}}$
7. ${\displaystyle {\sqrt {98}}}$
8. ${\displaystyle {\sqrt {75}}}$
9. ${\displaystyle {\sqrt {32}}}$
10. ${\displaystyle {\sqrt {320}}}$
11. ${\displaystyle {\sqrt {256}}}$
12. ${\displaystyle {\sqrt {\frac {9}{36}}}}$
13. ${\displaystyle {\sqrt[{3}]{\frac {27}{125}}}}$
14. ${\displaystyle {\sqrt {\frac {162}{200}}}}$
15. ${\displaystyle {\sqrt {\frac {75}{3}}}}$
16. ${\displaystyle {\sqrt {\frac {112}{72}}}}$

Rationalize the Denominator

17. ${\displaystyle {\frac {2}{\sqrt {4}}}}$
18. ${\displaystyle {\frac {8}{4+{\sqrt {26}}}}}$
19. ${\displaystyle {\frac {12}{12-{\sqrt {76}}}}}$
20. ${\displaystyle {\frac {3}{3-{\sqrt {65}}}}}$
21. ${\displaystyle {\frac {9}{9+{\sqrt {9}}}}}$
22. ${\displaystyle {\frac {a^{2}}{a+{\sqrt {a}}}}}$