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

These are the solutions to [[../Problems|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. $\frac{a^2-a\sqrt{a}}{a-1}$

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