Rational Exponents in Mathematics
Definition and Concept
Rational exponents, also known as fractional exponents, are an extension of the concept of integer exponents. They allow for a more nuanced expression of powers, particularly when dealing with roots and fractional powers. A rational exponent is expressed as $a^{m/n}$, where $a$ is the base, $m$ is the numerator, and $n$ is the denominator of the fractional exponent.
NoteThe expression $a^{m/n}$ is equivalent to $\sqrt[n]{a^m}$, which means taking the $n$th root of $a$ raised to the $m$th power.
Properties of Rational Exponents
- Product Rule: When multiplying expressions with the same base and rational exponents, add the exponents. $a^{m/n} \cdot a^{p/q} = a^{(m/n) + (p/q)}$
- Quotient Rule: When dividing expressions with the same base and rational exponents, subtract the exponents. $a^{m/n} \div a^{p/q} = a^{(m/n) - (p/q)}$
- Power Rule: When raising an expression with a rational exponent to another power, multiply the exponents. $(a^{m/n})^p = a^{(m/n)p}$
- Negative Exponents: A negative rational exponent indicates the reciprocal of the positive exponent. $a^{-m/n} = \frac{1}{a^{m/n}}$
Students often forget that $a^{-m/n}$ is not equal to $-a^{m/n}$. The negative sign affects the entire fraction, not just the numerator.
Simplifying Expressions with Rational Exponents
Simplifying expressions with rational exponents involves applying the properties mentioned above and often requires converting between radical and exponential forms.
ExampleLet's simplify $5^{1/2} \cdot 5^{1/3}$:
- Apply the product rule: $5^{1/2} \cdot 5^{1/3} = 5^{1/2 + 1/3}$
- Find a common denominator: $5^{3/6 + 2/6} = 5^{5/6}$
Therefore, $5^{1/2} \cdot 5^{1/3} = 5^{5/6}$
ExampleSimplify $6^{3/4} \div 6^{1/2}$:
- Apply the quotient rule: $6^{3/4} \div 6^{1/2} = 6^{3/4 - 1/2}$
- Find a common denominator: $6^{6/8 - 4/8} = 6^{2/8} = 6^{1/4}$
Thus, $6^{3/4} \div 6^{1/2} = 6^{1/4}$
