Find each product. Assume all variables represent positive real numbers. $-4k \left( k^{7/3} - 6k^{1/3} \right)$

Verified step by step guidance

1. Start by distributing the term outside the parentheses, which is \(-4k\), to each term inside the parentheses: \(-4k \times k^{7/3}\) and \(-4k \times (-6k^{1/3})\).
2. Recall the property of exponents when multiplying like bases: \(a^m \times a^n = a^{m+n}\). Use this to combine the powers of \(k\) in each product.
3. For the first product, add the exponents of \(k\): \$1\( (from \)k\() plus \)\frac{7}{3}\( (from \)k^{7/3}\(), so the exponent becomes \)1 + \frac{7}{3}$.
4. For the second product, multiply the constants \(-4\) and \(-6\) to get a positive product, then add the exponents of \(k\): \$1\( (from \)k\() plus \)\frac{1}{3}\( (from \)k^{1/3}\(), so the exponent becomes \)1 + \frac{1}{3}$.
5. Write the final expression as the sum of the two terms with their simplified coefficients and exponents, using the results from the previous steps.

Key Concepts

Properties of Exponents

Understanding how to manipulate exponents is essential, including the product rule which states that when multiplying like bases, you add their exponents. For example, k^a * k^b = k^(a+b). This helps simplify expressions involving variables raised to fractional powers.

Distributive Property

The distributive property allows you to multiply a single term across terms inside parentheses. For instance, a(b + c) = ab + ac. Applying this property correctly is crucial to expanding and simplifying algebraic expressions.

Fractional Exponents

Fractional exponents represent roots, such as k^(1/3) meaning the cube root of k. Recognizing and working with fractional exponents helps in simplifying expressions and combining like terms when variables have fractional powers.