- Factor the radicands by identifying perfect cube factors: \(128x^4 = 64x^3 \cdot 2x\) and \(16x^4 = 8x^3 \cdot 2x\).
- Simplify each radical: \(\sqrt[3]{128x^4} = 4x \sqrt[3]{2x}\) and \(\sqrt[3]{16x^4} = 2x \sqrt[3]{2x}\).
- Substitute these back into the original expression: \(-4x \sqrt[3]{2x} - 4(2x \sqrt[3]{2x})\).
- Combine the like terms to get the final result: \(\boxed{-12x \sqrt[3]{2x}}\).
Explanation
- Problem Analysis and Strategy
We are given the expression \(-\sqrt[3]{128 x^4} - 4 \sqrt[3]{16 x^4}\) where \(x > 0\). Our goal is to simplify this expression. We recall that the cube root of a product can be split using the property \(\sqrt[3]{a \cdot b} = \sqrt[3]{a} \cdot \sqrt[3]{b}\). Furthermore, for any non-negative \(a\), we know \(\sqrt[3]{a^3} = a\). Since \(x > 0\), we also have \(\sqrt[3]{x^3} = x\). We will simplify each term individually.
- Simplifying the First Term
First, consider the term \(-\sqrt[3]{128 x^4}\). We can factor the radicand \(128 x^4\) by looking for perfect cubes:
\(128 = 64 \times 2 = 4^3 \times 2\)
\(x^4 = x^3 \times x\)
Substituting these factors back into the cube root:
\(\sqrt[3]{128 x^4} = \sqrt[3]{64 \times 2 \times x^3 \times x} = \sqrt[3]{64} \times \sqrt[3]{x^3} \times \sqrt[3]{2x}\)
Since \(\sqrt[3]{64} = 4\) and \(\sqrt[3]{x^3} = x\), we get:
\(\sqrt[3]{128 x^4} = 4 \times x \times \sqrt[3]{2x} = 4x \sqrt[3]{2x}\)
Thus, the first term becomes \(-4x \sqrt[3]{2x}\).
- Simplifying the Second Term
Next, consider the second term \(-4 \sqrt[3]{16 x^4}\). We can factor the radicand \(16 x^4\) by looking for perfect cubes:
\(16 = 8 \times 2 = 2^3 \times 2\)
\(x^4 = x^3 \times x\)
Substituting these factors back into the cube root:
\(\sqrt[3]{16 x^4} = \sqrt[3]{8 \times 2 \times x^3 \times x} = \sqrt[3]{8} \times \sqrt[3]{x^3} \times \sqrt[3]{2x}\)
Since \(\sqrt[3]{8} = 2\) and \(\sqrt[3]{x^3} = x\), we get:
\(\sqrt[3]{16 x^4} = 2 \times x \times \sqrt[3]{2x} = 2x \sqrt[3]{2x}\)
The second term is therefore:
\(-4 \times (2x \sqrt[3]{2x}) = -8x \sqrt[3]{2x}\)
- Combining the Terms
Finally, we combine the two simplified terms:
\(-4x \sqrt[3]{2x} - 8x \sqrt[3]{2x} = (-4x - 8x) \sqrt[3]{2x} = -12x \sqrt[3]{2x}\)
Thus, the simplified form of the original expression is \(-12x \sqrt[3]{2x}\).
Examples
Simplifying algebraic expressions with radicals is essential in many fields, such as engineering and physics, where complex formulas need to be reduced for easier computation. For example, if you are modeling the volume of a sphere or the growth of a population where variables are raised to certain powers and then evaluated under roots, reducing these expressions to their simplest form helps in gaining better insights into how these quantities change. For instance, in thermodynamics, analyzing the relationship between heat capacity and temperature changes often requires simplifying radical terms to determine rate equations more efficiently.