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Precalculus: Radicals and Rational Exponents

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이 집합의 용어 (20)

  • Principal square root of a squared
    For any real number a, the principal square root of \(a^2\) is the absolute value of a.
  • Product rule for square roots
    If a and b are nonnegative real numbers, then \(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\).
  • Quotient rule for square roots
    If a and b are nonnegative real numbers and b ≠ 0, then \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\).
  • Like radicals
    Radicals that have the same radicand and index, which can be added or subtracted using the distributive property.
  • Rationalizing the denominator
    Rewriting a radical expression so the denominator contains no radicals by multiplying numerator and denominator by a suitable radical or conjugate.
  • Conjugates
    Expressions of the form \(a + \sqrt{b}\) and \(a - \sqrt{b}\); used to rationalize denominators with two terms.
  • Principal nth root of a real number
    For \(\sqrt[n]{a} = b\), if n is even, a and b are nonnegative; if n is odd, a and b can be any real numbers.
  • Radical and radicand
    The symbol \(\sqrt[n]{}\) is called a radical, and the expression under it is the radicand.
  • Finding nth roots of perfect nth powers (odd n)
    If n is odd, \(\sqrt[n]{a^n} = a\) for any real number a.
  • Finding nth roots of perfect nth powers (even n)
    If n is even, \(\sqrt[n]{a^n} = |a|\) where a is real.
  • Product rule for nth roots
    For real numbers a and b, \(\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}\).
  • Quotient rule for nth roots
    For real numbers a and b with b ≠ 0, \(\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}\).
  • Definition of a to the power of 1/n
    For real number a and positive integer n, \(a^{\frac{1}{n}} = \sqrt[n]{a}\).
  • Definition of a to the power of m/n
    For real number a, positive integer n, and integer m, \(a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}\).
  • Properties of rational exponents
    Rational exponents follow exponent rules: \(a^{m/n} \cdot a^{p/q} = a^{m/n + p/q}\) and \((a^{m/n})^{p/q} = a^{(m/n)(p/q)}\).
  • Simplifying expressions with rational exponents
    Use the definition of rational exponents and properties of exponents to rewrite and simplify expressions.
  • Adding and subtracting square roots
    Only like radicals (same radicand and index) can be added or subtracted by combining coefficients.
  • Rationalizing denominators with conjugates
    Multiply numerator and denominator by the conjugate of the denominator to eliminate radicals in denominators with two terms.
  • Simplifying square roots using product rule example
    Example: \(\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}\).
  • Simplifying square roots using quotient rule example
    Example: \(\frac{\sqrt{18}}{\sqrt{2}} = \sqrt{\frac{18}{2}} = \sqrt{9} = 3\).