Multiply by 5

Learning Objectives Express the number 5 using various rational exponents and bases. Multiply expressions containing rational exponents by 5 by correctly handling coefficients. Simplify expressions where multiplying by 5 involves combining terms with like bases and rational exponents. Apply the product rule of exponents when multiplying expressions where 5 can be expressed with a common base. Convert expressions between radical and rational exponent form after multiplication by 5. Solve equations involving rational exponents where a key step is to multiply by 5. You've been multiplying by 5 since elementary school, but what does it mean to multiply $x^{3/2}$ by 5? 🤔 Let's explore how this simple operation works in the complex world of rational exponents. This tut...

TermDefinitionExample Rational ExponentAn exponent expressed as a fraction m/n, where 'm' represents the power and 'n' represents the root. It follows the form $a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$.$27^{2/3} = (\sqrt[3]{27})^2 = 3^2 = 9$. CoefficientA numerical or constant quantity placed before and multiplying the variable in an algebraic expression.In the term $5x^{3/4}$, the coefficient is 5. BaseThe number or variable that is being raised to a power in an exponential expression.In the term $5x^{3/4}$, the base is x. Like TermsTerms that have the exact same base and the exact same exponent. Only like terms can be combined through addition or subtraction.$2a^{1/3}$ and $7a^{1/3}$ are like terms, but $2a^{1/3}$ and $7a^{1/2}$ are not. Radical FormA way of writin...

Multiplication by a Coefficient $c \cdot (a \cdot x^{m/n}) = (c \cdot a)x^{m/n}$ When multiplying a term with a rational exponent by a constant (like 5), multiply the constant by the term's existing coefficient. The base and the exponent do not change. Product Rule with a Common Base $b^p \cdot b^{q} = b^{p+q}$ Use this rule when the number you are multiplying by (e.g., 5) can be expressed with the same base as the term it is multiplying. To multiply, you add the exponents. For example, to calculate $5 \cdot 5^{1/2}$, you rewrite it as $5^1 \cdot 5^{1/2}$ and add the exponents to get $5^{3/2}$. Distributive Property $c(ax^{m/n} + by^{p/q}) = cax^{m/n} + cby^{p/q}$ When multiplying a polynomial by a constant (like 5), you must distribute the 5 to every term insid...