Formulas and properties of roots

Formulas and properties of roots are used to simplify and reduce complex expressions, and to solve equations and inequalities.

For a > 0 and b > 0 and natural numbers n, m, k the following relations hold:
  1. The root of a product of several factors equals the product of the roots of the same degree of these factors.
    $\sqrt[n]{a·b}=\sqrt[n]{a}·\sqrt[n]{b}$

  2. $\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}$

  3. $(\sqrt[n]{a})^k=\sqrt[n]{a^k}$

  4. $\sqrt[n]{\sqrt[m]{a}}=\sqrt[nm]{a}$

  5. $\sqrt[nk]{a^k}=\sqrt[n]{a}$

  6. $\sqrt[n·k]{a^{m·k}}=(\sqrt[n]{a^m})$

  7. $\sqrt[n]{a^n}=$ { |a|   if n is even
    a   if n is odd

  8. for any a and b such that 0 ≤ a ≤ b, the inequality holds:
    √ a  ≤ n√ b
  • Quick facts
  • Tables
  • Formulas
  • Geometry formulas
  • Math theory
  1. Sine values for 30, 45, 60 degrees.
  2. Geometry essentials for 8th grade. Module 1.
  3. Preparation for a geometry test in 8th grade.
  4. Inscribed angle theorem.
  5. Comparison of descent rates for a parachute jump and a jump from a window.
  1. Multiplication table (1 to 10).
  2. How to quickly learn the multiplication table.
  3. Extended multiplication table (1 to 20).
  4. Table of squares (1 to 10).
  5. Table of cubes (1 to 10).
  6. Table of powers (1 to 10).
  7. Table of factorials (1 to 10).
  8. Bradis table (with corrections).
  9. How to use the Bradis table. Visual guide.
  10. Table of sines.
  11. Table of cosines.
  12. Table of tangents.
  13. Table of cotangents.
  14. Table of prime numbers.
  15. Table of trigonometric functions.
  16. Table of function values.
  17. Table of natural logarithms.
  18. Table of decimal logarithms.
  19. Table of logarithms by base.
  1. Abbreviated multiplication formulas (2nd, 3rd, 4th and nth powers).
  2. Formulas and properties of powers.
  3. Formulas and properties of roots.
  4. Formulas and properties of logarithms.
  5. Formulas and properties of arithmetic progression.
  6. Formulas and properties of geometric progression.
  7. Trigonometric formulas.
  8. Inverse trigonometric functions.
  1. Areas of figures.
  2. Volumes of figures.
  3. Perimeters of figures.
  4. Surface areas of figures.
  5. Regular polygon.
  6. Triangle.
  7. Pythagorean theorem.
  1. Natural numbers.
  2. Gaussian method.
  3. Vieta's theorem.
  4. Fermat's Last Theorem.
  5. Number and digit.
  6. Divisibility of numbers.
  7. Decimal numbers.
  8. Classes of numbers.
  9. GCD and LCM.
  10. Rounding numbers.
  11. Prime and composite numbers.
  12. Place values of numbers.