Simplify radical expressions using the distributive property

Learning Objectives Apply the distributive property to expressions containing one or more radical terms. Multiply a monomial radical by a binomial or polynomial containing radicals. Multiply two binomial expressions containing radicals using the FOIL method. Simplify the products of radical expressions by combining like radical terms. Recognize and apply special product patterns, such as the difference of squares and perfect square trinomials, to radical expressions. Rationalize a denominator by multiplying by a conjugate, which is an application of the distributive property. Ever wondered how engineers calculate the precise dimensions of a structure with irrational lengths, like a bridge support with a length of (5 + √3) meters? 🤔 Let's find out! This tutorial will s...

TermDefinitionExample Radical ExpressionAn expression that contains a root symbol (√), such as a square root, cube root, or other higher root.3√5 + 2 Like RadicalsRadical terms that have the same index (the root being taken) and the same radicand (the value inside the root symbol).7√3 and -2√3 are like radicals. 7√3 and 7√5 are not. Distributive PropertyA property of algebra stating that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products.a(b + c) = ab + ac. Similarly, √2(5 + √3) = 5√2 + √2√3 = 5√2 + √6. FOIL MethodA mnemonic for multiplying two binomials: First, Outer, Inner, Last. It's a direct application of the distributive property twice.(a+b)(c+d) = ac + ad + bc + bd ConjugatesTwo binomials of the form (a + b) and (a -...

The Distributive Property with Radicals a√b (c√d + e√f) = ac√(bd) + ae√(bf) Use this rule to multiply a single radical term (monomial) across a sum or difference of other terms (polynomial). Multiply the coefficients together and multiply the radicands together. Product Rule for Radicals √a ⋅ √b = √(ab), where a ≥ 0 and b ≥ 0 When multiplying two radicals with the same index, you can combine them into a single radical by multiplying their radicands. This is essential for simplifying after distribution. Difference of Squares (Conjugates) (a + √b)(a - √b) = a² - (√b)² = a² - b When you multiply binomials that are conjugates, the middle terms cancel out, and the radical is eliminated. This is the key to rationalizing binomial denominators.