Multiply by 6

Learning Objectives Convert expressions between radical form (e.g., sixth roots) and rational exponent form. Simplify expressions with rational exponents where the base is 6, a multiple of 6, or a power of 6. Apply the product, quotient, and power rules to multiply and divide expressions with rational exponents involving the number 6. Evaluate complex numerical expressions containing rational exponents with a denominator of 6. Solve equations containing variables raised to rational powers, particularly when related to the number 6. Simplify algebraic expressions involving variables raised to fractional powers that include the number 6 in the numerator or denominator. Ever wonder how the simple act of 'multiplying by 6' evolves in advanced algebra? 🤯 Let's exp...

TermDefinitionExample Rational ExponentAn exponent expressed as a fraction m/n, where m is the power and n is the root. The general form is a^(m/n) = (ⁿ√a)ᵐ = ⁿ√(aᵐ).In the expression 64^(5/6), the rational exponent is 5/6. This means taking the sixth root of 64 and then raising it to the 5th power: (⁶√64)⁵ = 2⁵ = 32. Radical FormA way of writing an expression using the radical symbol (√) to indicate a root.The rational exponent expression x^(1/6) is written in radical form as ⁶√x. Principal Sixth RootFor a non-negative number 'a', its principal sixth root is the unique non-negative real number 'b' such that b⁶ = a.The principal sixth root of 64 is 2, because 2 is non-negative and 2⁶ = 64. BaseThe number or variable that is being raised to a power in an exponential exp...

Product of Powers Rule x^a ⋅ x^b = x^(a+b) When multiplying two exponential expressions with the same base, you add their exponents. This is fundamental for combining terms. For example, 6^(1/2) ⋅ 6^(1/3) = 6^(1/2 + 1/3) = 6^(5/6). Power of a Power Rule (x^a)^b = x^(a⋅b) When raising an exponential expression to another power, you multiply the exponents. This is key for simplifying nested exponents. For example, (36^(1/3))^3 = 36^(1/3 ⋅ 3) = 36^1 = 36. Power of a Product Rule (x ⋅ y)^a = x^a ⋅ y^a When a product is raised to a power, you can distribute the exponent to each factor in the product. For example, (6x)^(1/2) = 6^(1/2) ⋅ x^(1/2).