Questions: Laws of indices
Before attempting these questions, it is highly recommended that you read Guide: Laws of indices.
Q1
Express each of the following as a single real number.
1.1. \(\quad 3^4\)
1.2. \(\quad 125^{\frac{2}{3}}\)
1.3. \(\quad 32^{\frac{2}{5}}\)
1.4. \(\quad 729^{-\frac{2}{3}}\)
1.5. \(\quad 4^3\cdot2^5\)
1.6. \(\quad 2^2\cdot3^2\)
1.7. \(\quad 8^5\cdot6^5\)
1.8. \(\quad 12^{6}\cdot3^{6}\)
1.9. \(\displaystyle\quad \frac{9^2}{27^2}\)
1.10. \(\quad (5^2)^2\)
1.11. \(\quad (35^0)^9\)
1.12. \(\quad (35^9)^0\)
1.13. \(\quad (729^9)^{\frac{1}{9}}\)
1.14. \(\quad 7^{-3}\)
1.15. \(\quad \dfrac{4^5}{2^5}\)
1.16. \(\quad \dfrac{2^{-2}}{13^{-2}}\)
1.17. \(\quad 64^{\frac{4}{3}}\)
1.18. \(\displaystyle\quad \left(\frac{4^3\cdot{3^3}}{6^3}\right)\)
1.19. \(\quad \left(\dfrac{4^2\cdot{8^2}}{2^2}\right)\cdot{\left(\dfrac{1}{2}\right)^2}\)
1.20. \(\quad \dfrac{\left[\left(\frac{-2}{3}\right)^{-3}\cdot\left(\frac{-3}{5}\right)^{-3}\right]}{\left({\frac{2}{3}}\right)^{-3}}\)
1.21. \(\quad \dfrac{\left(\frac{1}{2}\right)^4\left(\frac{3}{5}\right)^4}{\left(\frac{8}{3}\right)^{4}}\)
1.22. \(\displaystyle\quad \left(\frac{2}{3}\right)^{14}\cdot\left(\frac{9}{12}\right)^{14}\)
Q2
Evaluate the following expressions, writing your answer in the simplest possible form.
2.1. \(\quad (b^{7})^{4}\)
2.2. \(\quad y^{13}\cdot{y^{5}}\)
2.3. \(\quad a^2\cdot b^2\)
2.4. \(\quad \dfrac{x^{13}}{x^5}\)
2.5. \(\quad (y^{-2})^5\)
2.6. \(\quad a^{-4}\cdot b^{-4}\)
2.7. \(\quad (7z^{-5})^3\)
2.8. \(\quad \dfrac{8x^5}{4x^{-5}}\)
2.9. \(\quad (x^{2})^3\cdot{x^{5}}\)
2.10. \(\quad \dfrac{2a^{-4}}{3a^{-2}}\)
2.11. \(\quad \dfrac{x^5}{y^5}\)
2.12. \(\quad \dfrac{2y^3}{2y^5}\)
2.13. \(\displaystyle\quad \left(\frac{2}{a}\right)^{4}\cdot\left(\frac{a}{12}\right)^{3}\)
2.14. \(\quad \dfrac{25t^{-4}}{60t^{5}}\)
2.15. \(\displaystyle\quad \left(\frac{a}{b}\right)^{-4}\cdot\left(\frac{c}{d}\right)^4\cdot\left(\frac{e}{f}\right)^4\)
2.16. \(\quad \dfrac{5^{x+1}\cdot6^{x+1}}{3^{x+1}}\)
2.17. \(\quad \left(a^{\frac{1}{2}}\right)\cdot\left(b^{-\frac{1}{2}}\right)\)
2.18. \(\displaystyle\quad \left(\frac{a}{b}\right)^n\cdot\left(\frac{c}{d}\right)^{-n}\)
After attempting the questions above, please click this link to find the answers..
Version history and licensing
v1.0: initial version created 08/23 by Isabella Lewis, Akshat Srivastava as part of a University of St Andrews STEP project.
- v1.1: edited 05/24 by tdhc.