What are the properties of rational exponents and how are they used to solve problems?
The properties of rational exponents are:
1. Product Rule: For any real numbers a and b, and any rational numbers m/n and p/q, where m, n, p, and q are integers:
- a^(m/n) * a^(p/q) = a^((m/n) + (p/q))
2. Quotient Rule: For any real numbers a and b (where a doesn't equal zero), and any rational numbers m/n and p/q, where m, n, p, and q are integers:
- (a^(m/n)) / (a^(p/q)) = a^((m/n) - (p/q))
3. Power Rule: For any real number a and any rational numbers m/n and p/q, where m, n, p, and q are integers:
- (a^(m/n))^((p/q)) = a^((m/n) * (p/q))
4. Zero Rule: For any rational number m/n, where m and n are integers and m isn't equal to zero:
- a^(m/n) = 0 if and only if a = 0
5. Identity Rule: For any rational number m/n, where m and n are integers:
- a^(m/n) = 1 if and only if a = 1, or m = 0
These properties are used to simplify expressions with rational exponents, perform operations involving rational exponents, and solve problems involving equations with rational exponents. By applying these properties, you can manipulate the exponents to combine terms, simplify expressions, or isolate variables.