Elementary Algebra

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Operations with Roots

Real numbers: \(a\), \(b\), \(p\)
Natural numbers: \(n\), \(m\)
  1. The \(n\)th root of a number \(a\) is called a number \(b,\) the \(n\)th power of which equals \(a.\) Here \(a\) and \(b\) are real numbers, \(n\) is a natural number \(\left({n \ge 2}\right).\)
    \(\sqrt[\large n\normalsize]{a} = b,\;\;{b^n} = a\)
  2. The principal or arithmetic \(n\)th root of a nonnegative number \(a\) is called a nonnegative number \(b,\) the \(n\)th power of which equals \(a.\) If \(a = 0,\) the principal \(n\)th root is also zero:
    \(\sqrt[\large n\normalsize]{0} = {0^{1/n}} = 0\)
  3. If \(a \lt 0,\) the \(n\)th root of a number \(a\) is defined only for odd indices n.
  4. The square root of a number \(a\) \(\left(a \ge 0\right)\) is usually denoted as \(\sqrt a \).
  5. \(N\)th root of a product \(\sqrt[\large n\normalsize]{{ab}} = \sqrt[\large n\normalsize]{a}\sqrt[\large n\normalsize]{b}\)
  6. Multiplication of roots with different bases and different indices
    \(\sqrt[\large n\normalsize]{a}\sqrt[\large m\normalsize]{b} = \sqrt[{\large nm\normalsize}]{{{a^m}{b^n}}}\)
  7. \(N\)th root of a quotient
    \(\sqrt[\large n\normalsize]{{\large\frac{a}{b}}}\normalsize = \large\frac{{\sqrt[n]{a}}}{{\sqrt[n]{b}}}\normalsize\;\;\left( {b \ne 0} \right)\)
  8. Division of roots with different bases and different indices
    \({\large\frac{{\sqrt[n]{a}}}{{\sqrt[m]{b}}}\normalsize} =\) \({\large\frac{{\sqrt[{nm}]{{{a^m}}}}}{{\sqrt[{nm}]{{{b^n}}}}}\normalsize} =\) \({\large\sqrt[{nm}]{{\frac{{{a^m}}}{{{b^n}}}}}\normalsize}\;\) \(\left( {b \ne 0} \right)\)
  9. Raising a root to a power \({\left( {\sqrt[\large n\normalsize]{a}} \right)^m} = \sqrt[\large n\normalsize]{{{a^m}}}\)
  10. \({\left( {\sqrt[\large n\normalsize]{a}} \right)^n} = a\)
  11. \(N\)th root of a power \(\sqrt[\large n\normalsize]{{{a^m}}} = {a^{m/n}}\)
  12. \(\sqrt[\large n\normalsize]{{{a^m}}} = \sqrt[{\large np\normalsize}]{{{a^{mp}}}}\)
  13. \({\left( {\sqrt[\large n\normalsize]{{{a^m}}}} \right)^p} = \sqrt[\large n\normalsize]{{{a^{mp}}}}\)
  14. Root of a root \(\sqrt[\large m\normalsize]{{\sqrt[\large n\normalsize]{a}}} = \sqrt[{\large mn\normalsize}]{a}\)
  15. Reciprocal of a root
    \({\large\frac{1}{{\sqrt[n]{a}}}\normalsize} = {\large\frac{{\sqrt[n]{{{a^{n – 1}}}}}}{a}\normalsize}\;\) \(\left( {a \ne 0} \right)\)
  16. \(\sqrt {a \pm \sqrt b } =\) \( \sqrt {\large\frac{{a + \sqrt {{a^2} – b} }}{2}\normalsize} \pm \sqrt {\large\frac{{a – \sqrt {{a^2} – b} }}{2}\normalsize} \;\) \(\left( {b \ge 0,a \ge \sqrt b } \right)\)
  17. Simplifying a radical expression
    \(\sqrt {a + \sqrt b } \pm \sqrt {a – \sqrt b } =\) \( 2\sqrt {\large\frac{{a \pm \sqrt {{a^2} – b} }}{2}\normalsize} \;\) \(\left( {b \ge 0,a \ge \sqrt b } \right)\)
  18. Rationalizing denominators
    \({\large\frac{1}{{\sqrt a \pm \sqrt b }}\normalsize} = {\large\frac{{\sqrt a \mp \sqrt b }}{{a – b}}\normalsize}\;\) \(\left( {a \ne b} \right)\)