Powers of monomials

Learning Objectives Define what a monomial is and identify its components. Apply the 'Power of a Power' rule to simplify expressions. Apply the 'Power of a Product' rule to simplify expressions. Simplify complex expressions involving powers of monomials with numerical coefficients and multiple variables. Correctly evaluate numerical coefficients when raising a monomial to a power. Identify and correct common errors made when simplifying powers of monomials. Ever wonder how engineers calculate the strength of materials or how scientists model the growth of populations? 📈 It often involves simplifying expressions with exponents! In this lesson, you'll learn how to simplify expressions where an entire monomial is raised to a power. Understanding thes...

TermDefinitionExample MonomialAn algebraic expression consisting of a single term, which can be a number, a variable, or a product of numbers and variables with whole number exponents.$5x^2y^3$, $-7a$, $12$, $z^4$ BaseIn an exponential expression $a^n$, the base is the number or variable that is being multiplied by itself.In $x^5$, $x$ is the base. In $(2y)^3$, $2y$ is the base. ExponentIn an exponential expression $a^n$, the exponent (or power) is the small number written above and to the right of the base, indicating how many times the base is multiplied by itself.In $x^5$, $5$ is the exponent. In $(2y)^3$, $3$ is the exponent. CoefficientThe numerical factor of a term that contains variables. If no number is written, the coefficient is 1.In $5x^2$, $5$ is the coefficient. In $-y^3$, $-...

Power of a Power Rule $(a^m)^n = a^{m \cdot n}$ When raising a power to another power, keep the base and multiply the exponents. This applies to variables within a monomial. Power of a Product Rule $(ab)^n = a^n b^n$ When raising a product to a power, raise each factor in the product to that power. This applies to both coefficients and variables within a monomial. Power of a Monomial Rule (Combined) $(ax^m)^n = a^n x^{m \cdot n}$ To raise a monomial to a power, raise the coefficient to that power and multiply the exponent of each variable by that power. This combines the Power of a Product and Power of a Power rules.