Exercises

Precalculus for Everybody

Chapter 1. Preliminaries

Radicals and Rational Exponents

In the exercises, from 1 to 9, evaluate the expression

  1. \[ \sqrt{(-5)^2} \]
  2. \[ \sqrt[3]{-0.027} \]
  3. \[ (0.16)^{-1/2} \]
  4. \[ (32)^{-2/5} \]
  5. \[ \left( -\frac{8}{27} \right)^{-1/3} \]
  6. \[ (0.0016)^{-3.4} \]
  7. \[ 5^{2/7}5^{5/7} \]
  8. \[ (125)^{-2/3} \div (81)^{1/4} \]
  9. \[ \left[ (243)^{-4/5}(64)^{2/3} \right]^{1/4} \]

In the exercises, from 10 to 14, simplify the expression. Do not use negative exponents in the final answer

  1. \[ \left(-2a^{-3}b \right)^2 \left(3a^2b^{-1} \right)^3 \]
  2. \[ \left( \frac{ 3x^2 }{ y^3 } \right)^2 \left( \frac{ -2x^2 }{ 3y } \right)^{-2} \]
  3. \[ \frac{ \left( x^{-3} y^2 \right)^3 }{ \left( x^3 y^{-2} \right)^2 } \]
  4. \[ \frac{ \left( 32 a^{15} c^{-5} \right)^{1/5} }{ \left( -27 a^6 c^{-3} \right)^{1/3} } \]
  5. \[ \left( \frac{ x^{-2} y^3 }{ x^4 y^{-3} } \right)^{- 1/2} \left( \frac{ x^4 y^{-4} }{ x y^2 } \right)^{- 1/3} \]

In the exercises, from 15 to 25, simplify expression. Rationalize the denominators if necessary.

  1. \[ 5\sqrt{20}-3\sqrt{45}+\frac{\sqrt{80}}{2} \]
  2. \[ \sqrt{243}-\sqrt{63}+\sqrt{175}-2\sqrt{75} \]
  3. \[ \frac{\sqrt{48}+\sqrt{75}}{-\sqrt{81}} \]
  4. \[ \frac{\sqrt{2}}{\sqrt{72}-\sqrt{8}+\sqrt{50}} \]
  5. \[ \sqrt[3]{1,080} - \sqrt[3]{625} + \sqrt[3]{40} \]
  6. \[ \sqrt[3]{-375} - \sqrt[3]{-24} - 4 \sqrt[3]{-81} \]
  7. \[ \frac{56}{\sqrt{7}}-6\sqrt{28}+\frac{\sqrt{343}}{7} \]
  8. \[ \sqrt{75}-3\sqrt{\frac{4}{3}}+\sqrt{48} \]
  9. \[ \sqrt{\frac{3}{8}}-\sqrt{\frac{2}{3}}-\frac{\sqrt{24}}{3} \]
  10. \[ \sqrt{\frac{1}{12}}-\sqrt{\frac{1}{3}}+\sqrt{\frac{3}{4}} \]
  11. \[ \sqrt[3]{ \frac{1}{4} } + \sqrt[3]{ \frac{1}{32} } - \sqrt[3]{ \frac{2}{27} } \]

In the exercises 26 and 27, simplify.

  1. \[ \frac{ 2^{n - 2} - 2^{n - 1} + 2^n }{ 2^{n + 2} - 2^{n + 1} + 2^n } \]
  2. \[ \frac{ 12^n \times 225^{n / 2} \times 35^{2n} }{ 49^n \times 16^{n/4} \times 27^{ 2n/3 } } \]

In the exercises, from 28 to 30, find the value of \(\boldsymbol{n}\).

  1. \[ 5\sqrt{5}\sqrt[3]{25}=5^n \]
  2. \[ \sqrt{\sqrt[5]{3}}=3^n \]
  3. \[ \sqrt[n]{ \sqrt[n]{ 5 } } = 5^{1/9} \]
  1. \[ 5 \]
     
  2. \[ -0.3 \]
     
  3. \[ \frac{1}{0.4}=\frac{5}{2} \]
     
  4. \[ \frac{1}{2^2} = \frac{1}{4} \]
     
  5. \[ -\frac{3}{2} \]
     
  6. \[ 125 \]
     
  7. \[ 5 \]
     
  8. \[ \frac{1}{75} \]
     
  9. \[ \frac{2}{3} \]
     
  10. \[ \frac{108}{b} \]
     
  11. \[ \frac{81}{4y^4} \]
     
  12. \[ \frac{y^{10}}{x^{15}} \]
     
  13. \[ -\frac{2a}{3} \]
     
  14. \[ \frac{x^2}{y} \]
     
  15. \[ 3\sqrt{5} \]
     
  16. \[ 2 \sqrt{7}- \sqrt{3} \]
     
  17. \[ -\sqrt{3} \]
     
  18. \[ \frac{1}{9} \]
     
  19. \[ 3 \sqrt[3]{5} \]
     
  20. \[ 13 \sqrt{3} \]
     
  21. \[ – 3\sqrt{7} \]
     
  22. \[ 7 \sqrt{3} \]
     
  23. \[ -\frac{3}{4}\sqrt{6} \]
     
  24. \[ \frac{\sqrt{3}}{3} \]
     
  25. \[ \frac{5}{12} \sqrt[3]{2} \]
     
  26. \[ \frac{1}{4} \]
     
  27. \[ 2^n \times 5^{3n} = 250^n \]
     
  28. \[ n = \frac{13}{6} \]
     
  29. \[ n = \frac{1}{10} \]
     
  30. \[ n = 3 \]