Operations with Rational Exponents
Operations with rational exponents is a Grade 7 math skill from Yoshiwara Intermediate Algebra applying standard exponent rules to fractional exponents. Students simplify products, quotients, and powers of expressions like x^(2/3) or x^(1/2) · x^(3/2) using the laws of exponents.
Key Concepts
Property Powers with rational exponents obey the same laws of exponents as powers with integer exponents. For a base $a 0$ and rational exponents $p$ and $q$:.
1. First Law (Product of Powers): $a^p \cdot a^q = a^{p+q}$ 2. Second Law (Quotient of Powers): $\frac{a^p}{a^q} = a^{p q}$ 3. Third Law (Power of a Power): $(a^p)^q = a^{pq}$ 4. Fourth Law (Power of a Product): $(ab)^p = a^p b^p$.
Examples To simplify $x^{1/2} \cdot x^{1/4}$, we add the exponents: $x^{1/2 + 1/4} = x^{2/4 + 1/4} = x^{3/4}$.
Common Questions
What are the rules for multiplying rational exponents?
When multiplying with the same base, add the exponents: x^(m/n) · x^(p/q) = x^(m/n + p/q).
How do you simplify x^(1/2) · x^(3/2)?
Add exponents: x^(1/2 + 3/2) = x^(4/2) = x^2.
How do you simplify (x^(2/3))^3?
Multiply exponents: x^(2/3 · 3) = x^2.
How do you divide rational exponents?
Subtract exponents when dividing: x^(5/3) ÷ x^(2/3) = x^(5/3 - 2/3) = x^1 = x.