Multiplication with rational exponents

Learning Objectives Apply the product rule to multiply expressions with rational exponents and the same base. Simplify expressions involving multiplication of terms with different bases but convertible to a common base. Correctly add fractional exponents by finding a common denominator. Multiply expressions containing both numerical coefficients and variables with rational exponents. Convert between radical and exponential forms to simplify multiplication problems. Apply the power of a product rule in conjunction with the product rule for exponents. How can a scientist calculate the rapid growth of a bacterial colony over fractions of an hour? 🔬 It all comes down to understanding multiplication with fractional exponents! This tutorial focuses on the rules for multiplying e...

TermDefinitionExample Rational ExponentAn exponent that is a fraction, in the form p/q, where p is the power and q is the root.In x^(2/3), 2 is the power and 3 is the cube root. This is equivalent to (∛x)². BaseThe number or variable that is being raised to a power.In 5^(3/4), the base is 5. Product Rule for ExponentsWhen multiplying two exponential expressions with the same base, you keep the base and add the exponents.x^(1/2) * x^(1/3) = x^(1/2 + 1/3) = x^(5/6) Common DenominatorA shared multiple of the denominators of several fractions. It is required to add or subtract fractions.To add 1/2 and 1/3, the common denominator is 6. So, 3/6 + 2/6 = 5/6. CoefficientA numerical factor that multiplies a variable.In the term 7x^(1/2), the coefficient is 7.

Product Rule (Same Base) x^{a} \cdot x^{b} = x^{a+b} Use this rule when you are multiplying two or more terms that have the exact same base. Keep the base the same and add the rational exponents together. Power of a Product Rule (xy)^{a} = x^{a} \cdot y^{a} Use this rule to distribute an outer exponent to each factor inside a set of parentheses. This is often a first step before applying the Product Rule.