Perform the indicated operations and/or simplify each expression. Assume all variables represent positive real numbers. ∛2/3

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Identify the expression given: the cube root of the fraction $ \frac{2}{3} $, which can be written as $ \sqrt[3]{\frac{2}{3}} $.

Recall the property of radicals that allows you to separate the cube root of a fraction into the cube root of the numerator divided by the cube root of the denominator: $ \sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}} $.

Apply this property to rewrite the expression as $ \frac{\sqrt[3]{2}}{\sqrt[3]{3}} $.

Since the problem asks to simplify, consider rationalizing the denominator by multiplying numerator and denominator by $ \sqrt[3]{3^2} = \sqrt[3]{9} $ to eliminate the cube root in the denominator.

Perform the multiplication: $ \frac{\sqrt[3]{2} \times \sqrt[3]{9}}{\sqrt[3]{3} \times \sqrt[3]{9}} = \frac{\sqrt[3]{18}}{\sqrt[3]{27}} $, and then simplify the denominator $ \sqrt[3]{27} $ since 27 is a perfect cube.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Cube Roots

A cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, ∛8 = 2 because 2³ = 8. Understanding cube roots helps simplify expressions involving radicals with an index of 3.

Radical Expressions and Simplification

Radical expressions involve roots such as square roots or cube roots. Simplifying these expressions means rewriting them in the simplest form, often by factoring the radicand or rationalizing denominators, to make calculations easier and expressions clearer.

Properties of Exponents and Radicals

Radicals can be expressed using fractional exponents, where the nth root of a number is the same as raising it to the 1/n power. For example, ∛(2/3) = (2/3)^(1/3). Using exponent rules helps in performing operations and simplifying radical expressions.

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Rational Exponents

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Factor out the least power of the variable or variable expression. Assume all variables represent positive real numbers. See Example 8.

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Factor out the least power of the variable or variable expression. Assume all variables represent positive real numbers. See Example 8. $6y^3(4y - 1)^{-3/7} - 8y^2(4y - 1)^{4/7} + 16y(4y - 1)^{11/7}$