Perform the indicated operations and/or simplify each expression. Assume all variables represent positive real numbers. ∛(8/x⁴)

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Identify the expression given: \(\sqrt[3]{\frac{8}{x^4}}\) which means the cube root of the fraction \(\frac{8}{x^4}\).

Recall that the cube root of a fraction can be written as the fraction of the cube roots: \(\sqrt[3]{\frac{8}{x^4}} = \frac{\sqrt[3]{8}}{\sqrt[3]{x^4}}\).

Simplify the cube root of the numerator: since \(8 = 2^3\), \(\sqrt[3]{8} = 2\).

Rewrite the cube root of the denominator using exponent rules: \(\sqrt[3]{x^4} = x^{\frac{4}{3}}\) because the cube root is the same as raising to the power \(\frac{1}{3}\).

Combine the simplified numerator and denominator to write the expression as \(\frac{2}{x^{\frac{4}{3}}}\), which is the simplified form.

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Cube Roots and Radicals

A cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, ∛8 equals 2 because 2³ = 8. Understanding how to simplify cube roots is essential for manipulating expressions involving radicals.

Properties of Exponents

Exponents indicate repeated multiplication of a base number. Key properties include the product rule, quotient rule, and power rule, which help simplify expressions with variables raised to powers. For example, x⁴ means x multiplied by itself four times.

Simplifying Algebraic Fractions

Simplifying algebraic fractions involves reducing expressions by factoring, canceling common terms, and applying exponent rules. When variables are in denominators with exponents, rewriting them using negative exponents or radicals can aid simplification.