Simplify each radical. Assume all variables represent positive real numbers. ∛(16 (-2)⁴ (2)⁸)

Verified step by step guidance

1. Start by rewriting the expression inside the cube root to clearly see all factors: \(\sqrt[3]{16 \cdot (-2)^4 \cdot 2^8}\).
2. Simplify each base raised to a power: calculate \((-2)^4\) and keep \$2^8\( as is. Remember that \)(-2)^4$ means \((-2) \times (-2) \times (-2) \times (-2)\).
3. Express all numbers as powers of 2 to combine them easily. For example, write 16 as \$2^4\( and simplify \)(-2)^4$ as a power of 2, considering the sign and exponent.
4. Combine all powers of 2 by adding their exponents inside the cube root, since multiplication of like bases corresponds to adding exponents: \(2^{a} \cdot 2^{b} = 2^{a+b}\).
5. Apply the cube root to the combined power of 2 by dividing the exponent by 3, using the property \(\sqrt[3]{2^{k}} = 2^{k/3}\). This will give the simplified radical expression.

Key Concepts

Properties of Exponents

Understanding how to manipulate exponents is essential for simplifying expressions involving powers. This includes applying rules such as multiplying powers with the same base by adding exponents and raising a power to another power by multiplying exponents.

Simplifying Radicals (Cube Roots)

Simplifying cube roots involves expressing the radicand as a product of perfect cubes and other factors. Extracting cube roots of perfect cubes simplifies the expression, while remaining factors stay under the radical.

Assumption of Positive Variables

Assuming all variables represent positive real numbers allows us to simplify expressions without considering absolute values or negative roots. This assumption ensures the principal root is taken and simplifies the handling of variables under radicals.