Find each product. Assume all variables represent positive real numbers. $y^{5/8} \left( y^{3/8} - 10y^{11/8} \right)$

Verified step by step guidance

1. Identify the expression to be simplified: \(y^{\frac{5}{8}} \left(y^{\frac{3}{8}} - 10y^{\frac{11}{8}}\right)\).
2. Apply the distributive property by multiplying \(y^{\frac{5}{8}}\) with each term inside the parentheses separately: \(y^{\frac{5}{8}} \cdot y^{\frac{3}{8}}\) and \(y^{\frac{5}{8}} \cdot (-10y^{\frac{11}{8}})\).
3. Use the rule of exponents for multiplying powers with the same base: \(a^m \cdot a^n = a^{m+n}\). So, add the exponents when multiplying the \(y\) terms.
4. Calculate the exponents for each product: For the first term, add \(\frac{5}{8} + \frac{3}{8}\); for the second term, add \(\frac{5}{8} + \frac{11}{8}\).
5. Rewrite the expression with the new exponents and simplify the coefficients: \(y^{\text{sum of exponents}}\) for each term, keeping the \(-10\) coefficient in the second term.

Key Concepts

Laws of Exponents

The laws of exponents govern how to multiply, divide, and simplify expressions with powers. When multiplying terms with the same base, add their exponents. For example, y^(5/8) * y^(3/8) = y^((5/8)+(3/8)) = y^(8/8) = y^1. Understanding these rules is essential for simplifying the given expression.

Distributive Property

The distributive property states that a(b + c) = ab + ac. It allows you to multiply a single term by each term inside a parenthesis. In this problem, y^(5/8) must be multiplied by both y^(3/8) and -10y^(11/8), distributing the multiplication across the terms.

Simplifying Algebraic Expressions

Simplifying involves combining like terms and reducing expressions to their simplest form. After applying exponent rules and distribution, combine powers and constants carefully. Since all variables are positive, no absolute values or domain restrictions affect simplification.