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

Verified step by step guidance

1. Start by rewriting the expression inside the cube root: \(\sqrt[3]{25 \cdot (-3)^4 \cdot 5^3}\).
2. Calculate the powers inside the radical separately: \((-3)^4\) means \((-3) \times (-3) \times (-3) \times (-3)\), and \$5^3\( means \)5 \times 5 \times 5$.
3. Express all numbers as products of prime factors or powers to identify perfect cubes. For example, write 25 as \$5^2\(, and use the results from the previous step for \)(-3)^4\( and \)5^3$.
4. Combine all factors inside the cube root, grouping the powers of the same base together: \(\sqrt[3]{5^2 \cdot (-3)^4 \cdot 5^3} = \sqrt[3]{5^{2+3} \cdot (-3)^4}\).
5. Use the property \(\sqrt[3]{a^b} = a^{\frac{b}{3}}\) to simplify each factor by dividing the exponents by 3, separating the parts that are perfect cubes (exponents divisible by 3) from the leftover parts inside the cube root.

Key Concepts

Properties of Exponents

Understanding how to manipulate exponents is essential for simplifying expressions involving powers. This includes knowing how to multiply powers with the same base by adding exponents and raising a power to another power by multiplying exponents. These rules help in rewriting terms inside the radical for easier simplification.

Simplifying Radicals (Cube Roots)

Simplifying cube roots involves expressing the radicand as a product of perfect cubes and other factors. The cube root of a perfect cube is an integer, which can be taken outside the radical. Recognizing and extracting these perfect cubes simplifies the expression significantly.

Assumption of Positive Variables

Assuming all variables represent positive real numbers allows us to simplify radicals without considering absolute values. This assumption ensures that the principal root is positive, which simplifies the process and avoids ambiguity in the final simplified expression.