Laws of Exponents with Rational Exponents
Rational Exponents and Roots
For any positive real number $a$ and rational number $\frac{m}{n}$ where $m$ and $n$ are integers and $n \neq 0$:
$$a^{\frac{m}{n}} = \sqrt[n]{a^m}$$
ExampleLet's evaluate $8^{\frac{2}{3}}$:
$$8^{\frac{2}{3}} = \sqrt[3]{8^2} = \sqrt[3]{64} = 4$$
Here, we first square 8, then take the cube root of the result.
Note- When $n$ is even, $a^{\frac{1}{n}}$ refers specifically to the positive $n$th root of $a$.
- This convention ensures that exponential expressions with rational exponents are well-defined for positive real bases.
Properties of Exponents
- The laws of exponents are as follows:
- Product rule: $a^x \cdot a^y = a^{x+y}$
- Quotient rule: $\frac{a^x}{a^y} = a^{x-y}$
- Power rule: $(a^x)^y = a^{xy}$
- These rules apply regardless of whether $x$ and $y$ are integers, rational, or real numbers.
Evaluate $27^{\frac{4}{3}}$.
Solution
$$27^{\frac{4}{3}} = 27^{\frac{1}{3} \cdot 4} = \sqrt[3]{27}^4 = 3^4 = 81$$
We applied the power rule, then simplified the rational exponent, and finally evaluated the result.
- Students often confuse $a^{\frac{m}{n}}$ with $\frac{a^m}{a^n}$.
- Remember, $a^{\frac{m}{n}}$ is a single number raised to a fractional power, while $\frac{a^m}{a^n}$ is a quotient of two exponential expressions that evaluates to $a^{m - n}$.
Laws of Logarithms
Logarithms are the inverse operations of exponents, and they follow a set of laws that mirror the properties of exponents.
Basic Logarithm Laws
- The basic logarithm laws can be derived from the exponent laws.
- For positive real numbers $a$, $x$, and $y$, where $a \neq 1$:
- Product rule: $\log_a(xy) = \log_a(x) + \log_a(y)$
- Quotient rule: $\log_a(\frac{x}{y}) = \log_a(x) - \log_a(y)$
- Power rule: $\log_a(x^m) = m \log_a(x)$
Simplify $\log_2(16\sqrt2)$.
Solution
$$\log_2(16\sqrt2) = \log_2(16)+\log_2(\sqrt2) = 4+\frac12 = \frac92$$
We applied the product rule in reverse, then evaluated the logarithm.
- When simplifying logarithmic expressions, look for opportunities to apply these laws in either direction.
- Sometimes combining logarithms is helpful, while other times separating them is more useful.
