Study Notes on Rational Exponents

DEFINITION OF RATIONAL EXPONENTS

  • For any rational exponent $ rac{m}{n}$ in lowest terms, where $m$ and $n$ are integers and $n eq 0$, we define:
    • amn=(an)ma^\frac{m}{n} = (\sqrt[n]{a})^m
    • Equivalently, we may write that: amn=amna^\frac{m}{n} = \sqrt[n]{a^m}
  • If $n$ is even, then we require that a≥0a \geq 0.
  • This definition allows for the application of the Laws of Exponents to rational exponents.

EXAMPLES OF RATIONAL EXPONENTS

  • Example 11: Using the Definition of Rational Exponents
    • (a) 41/2=4=24^{1/2} = \sqrt{4} = 2
    • (b) 82/3=(8)2=48^{2/3} = (8)^2 = 4
    • Alternative solution: 82/3=82=6433=48^{2/3} = 8^{2} = \sqrt[3]{\sqrt[3]{64}} = 4
    • (c) 125−1/3=11251/3=11253=15125^{-1/3} = \frac{1}{125^{1/3}} = \frac{1}{\sqrt[3]{125}} = \frac{1}{5}

APPLICATION OF LAWS OF EXPONENTS WITH RATIONAL EXPONENTS

  • Example 12: Using the Laws of Exponents with Rational Exponents
    • (a) a3a7/3=a3+7/3a^3 a^{7/3} = a^{3 + 7/3}
    • (Using Law 1: $a^m a^n = a^{m + n}$)
    • (b) a2/5a7/5a3/5=a25+75−35=a65\frac{a^{2/5} a^{7/5}}{a^{3/5}} = a^{\frac{2}{5} + \frac{7}{5} - \frac{3}{5}} = a^{\frac{6}{5}}
    • (Applying Law 1 and Law 2: $a^{m/n} = a^{-n}$ )
    • (c) (2a3b4)3/2=23/2imes(a3)3/2imes(b4)3/2(2 a^3 b^4)^{3/2} = 2^{3/2} imes (a^3)^{3/2} imes (b^4)^{3/2}
    • =(2)3a3imes32b4imes32=22a9b6= (\sqrt{2})^3 a^{3 imes \frac{3}{2}} b^{4 imes \frac{3}{2}} = 2\sqrt{2} a^{9} b^{6}
    • (d) 2x3/4y1/3imes−1/2=2 x^{3/4} y^{1/3} imes - 1/2 =
    • =23(x3/4)3(y1/3)−1/2=8x9/4y1/6= 2^3 (x^{3/4})^{3} (y^{1/3})^{-1/2} = \frac{8 x^{9/4}}{y^{1/6}}

SIMPLIFYING BY WRITING RADICALS AS RATIONAL EXPONENTS

  • Example 13: Simplifying Expressions
    • (a) x−4/3=1x4/3x^{-4/3} = \frac{1}{x^{4/3}}
    • (Using the definition of rational and negative exponents)
    • (b) (2x)(3x)=(2x1/2)(3x1/3)(2\sqrt{x})(3x) = (2x^{1/2}) (3x^{1/3})
    • =6x1/2+1/3=6x56= 6x^{1/2 + 1/3} = 6x^{\frac{5}{6}}
    • (Law 1 applied here)
    • (c) xx=(x1/2)1/2=(x3/2)1/2=x3/4\sqrt{x} \sqrt{x} = (x^{1/2})^{1/2} = (x^{3/2})^{1/2} = x^{3/4}

RATIONALIZING THE DENOMINATOR

  • It is often useful to eliminate the radical in a denominator by multiplying both the numerator and denominator by an appropriate expression; this process is called rationalizing the denominator.
    • If the denominator is of the form a\sqrt{a}, we multiply the numerator and denominator by a\sqrt{a}:
    • For instance, 1a×aa=aa\frac{1}{\sqrt{a}} \times \frac{\sqrt{a}}{\sqrt{a}} = \frac{\sqrt{a}}{a}
    • The multiplication by1 (since aa\frac{\sqrt{a}}{\sqrt{a}} equals 1) doesn't change the value.
    • Note that in the last fraction, the denominator no longer contains a radical.
  • In general, if the denominator is of the form anm\sqrt[m]{a^n} with $m < n$, multiplying the numerator and denominator by an−mm\sqrt[m]{a^{n-m}} will rationalize the denominator because for $a > 0$:
    • amm⋅an−mm=anm=a\sqrt[m]{a^m} \cdot \sqrt[m]{a^{n-m}} = \sqrt[m]{a^{n}} = a