Simplify each radical. Assume all variables represent positive real numbers. ∛(25 (-3)⁴ (5)³ )

Verified step by step guidance

1. Identify the expression inside the cube root: \(25 (-3)^4 (5)^3\).
2. Simplify the powers: \((-3)^4 = 81\) and \((5)^3 = 125\).
3. Substitute the simplified powers back into the expression: \(25 \times 81 \times 125\).
4. Multiply the numbers inside the cube root: \(25 \times 81 = 2025\) and then \(2025 \times 125\).
5. Take the cube root of the resulting product to simplify the expression.

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. In this context, the cube root (∛) is used to simplify the expression. Understanding how to manipulate and simplify these expressions is crucial, especially when dealing with products and powers within the radical.

Properties of Exponents

The properties of exponents govern how to simplify expressions involving powers. Key rules include the product of powers (a^m * a^n = a^(m+n)) and the power of a power ( (a^m)^n = a^(m*n)). These rules are essential for simplifying the terms inside the radical before taking the cube root.

Simplifying Radicals

Simplifying radicals involves breaking down the expression into its prime factors and identifying perfect cubes (or squares, depending on the root). This process allows for the extraction of whole numbers from the radical, making the expression easier to work with. It is important to recognize how to factor the expression correctly to simplify it effectively.