Honors Algebra 2: Extra Practice on Simplifying Radicals



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Simplify the following. Express your answer in simplest radical form.

  1. \( \displaystyle \sqrt[4]{48} \)
  2. Solution

    \( \displaystyle 2\sqrt[4]{3} \)

  3. \( \displaystyle \sqrt{75x^3} \)
  4. Solution

    \( \displaystyle 5x\sqrt{3x}\)

  5. \( \displaystyle \sqrt[4]{\left(5x\right)^4} \)
  6. Solution

    \( \displaystyle 5|x|\)

  7. \( \displaystyle \sqrt[3]{24} \)
  8. Solution

    \( \displaystyle 2\sqrt[3]{3}\)

  9. \( \displaystyle \sqrt[3]{24a^4} \)
  10. Solution

    \( \displaystyle 2a\sqrt[3]{3a}\)

  11. \( \displaystyle \sqrt[3]{-40x^6} \)
  12. Solution

    \( \displaystyle -2x^2 \sqrt[3]{5} \)

  13. \( \displaystyle \frac{\sqrt{8}}{2} \)
  14. Solution

    \( \displaystyle \sqrt{2}\)

  15. \( \displaystyle \frac{\sqrt{5} + \sqrt{3}}{\sqrt{3}} \)
  16. Solution

    \( \displaystyle \frac{\sqrt{15} + 3}{3} \)

  17. \( \displaystyle \frac{\sqrt{12} - 3}{\sqrt{8}} \)
  18. Solution

    \( \displaystyle \frac{2\sqrt{6} - 3\sqrt{2}}{4} \)

  19. \( \displaystyle \sqrt{\sqrt[4]{2x}} \)
  20. Solution

    \( \displaystyle \sqrt[8]{2x} \)

  21. \( \displaystyle \sqrt[5]{\sqrt[3]{10a^8b^{15}}} \)
  22. Solution

    \( \displaystyle b\sqrt[15]{10a^8} \)

  23. \( \displaystyle \sqrt[3]{\frac{5}{4y^2}} \)
  24. Solution

    \( \displaystyle \frac{\sqrt[3]{10y}}{2y} \)

  25. \( \displaystyle \frac{6a^2}{\sqrt[4]{8a^2}} \)
  26. Solution

    \( \displaystyle 3|a|\sqrt[4]{2a^2} \)

  27. \( \displaystyle \frac{12n}{\sqrt[3]{9n^7}} \)
  28. Solution

    \( \displaystyle \frac{4\sqrt[3]{3n^2}}{n^2} \)

  29. \( \displaystyle \frac{4}{\sqrt[3]{7} - \sqrt[3]{5}} \)
  30. Solution

    \( \displaystyle 2\sqrt[3]{49} + 2\sqrt[3]{35} + 2\sqrt[3]{25} \)

  31. \( \displaystyle \frac{-2}{\sqrt[3]{a} + 4} \)
  32. Solution

    \( \displaystyle \frac{-2\sqrt[3]{a^2} + 8\sqrt[3]{a} - 32}{a + 64} \)

  33. \( \displaystyle \sqrt[3]{x^{6n}} \)
  34. Solution

    \( \displaystyle x^{2n} \)

  35. \( \displaystyle \sqrt[4]{y^{4n + 12}} \)
  36. Solution

    \( \displaystyle \left| y^{n + 3}\right| \)

  37. \( \displaystyle \sqrt[3]{x^{2n + 1}} \cdot \sqrt[3]{x^{4n + 2}} \)
  38. Solution

    \( \displaystyle x^{2n + 1}\)

  39. \( \displaystyle y^n \cdot \sqrt[4]{y^{3n}} + \sqrt[4]{y^{7n}}, y > 0. \)
  40. Solution

    \( \displaystyle 2y^n\sqrt[4]{y^{3n}} \)

  41. \( \displaystyle \sqrt[n]{x^n} \)
  42. Solution

    \( x \) if \(n\) is odd; \( |x| \) if \(n\) is even

  43. \( \displaystyle \sqrt[n]{3^{3n + 2}} \)
  44. Solution

    \( \displaystyle 27 \sqrt[n]{9} \)

  45. \( \displaystyle \sqrt[n]{a^nx^{5n+1}} \) for \(a, x > 0\)
  46. Solution

    \( \displaystyle ax^5 \sqrt[n]{x} \) for \(a, x > 0\)

  47. \( \displaystyle \sqrt[n]{ab^{2n}x^{2n+3}} \) for \(a, x > 0\)
  48. Solution

    \( \displaystyle b^2x^2\sqrt[n]{ax^3} \) for \(a, x > 0\)