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

Surds

Simplify the following Surds.

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

Rationalize the Denominator

1. ${\frac {2}{\sqrt {4}}}$
2. ${\frac {8}{4+{\sqrt {26}}}}$
3. ${\frac {12}{12-{\sqrt {76}}}}$
4. ${\frac {3}{3-{\sqrt {65}}}}$
5. ${\frac {9}{9+{\sqrt {9}}}}$
6. ${\frac {a^{2}}{a+{\sqrt {a}}}}$