- Simplify the individual radical terms by factoring out perfect cubes: \(-3\sqrt[3]{16x} = -6\sqrt[3]{2x}\) and \(-3\sqrt[3]{8x} = -6\sqrt[3]{x}\).
- Substitute the simplified terms back into the original expression to get \(7\sqrt[3]{2x} - 6\sqrt[3]{2x} - 6\sqrt[3]{x}\).
- Combine the like terms \(7\sqrt[3]{2x}\) and \(-6\sqrt[3]{2x}\) to result in \(\sqrt[3]{2x} - 6\sqrt[3]{x}\).
- The final simplified form is \(\boxed{\sqrt[3]{2x} - 6\sqrt[3]{x}}\).
Explanation
- Problem Analysis and Initial Approach
To simplify the expression \(7\sqrt[3]{2x} - 3\sqrt[3]{16x} - 3\sqrt[3]{8x}\), we analyze each term individually. The expression is composed of three cube root terms. We recall that the cube root of a product can be split into the product of the cube roots, specifically \(\sqrt[3]{ab} = \sqrt[3]{a} \cdot \sqrt[3]{b}\), and that the cube root of a perfect cube is \(\sqrt[3]{k^3} = k\).
- Simplifying Each Term
The first term is already in its simplest form, \(7\sqrt[3]{2x}\). For the second term, we observe that \(16x = 8 \cdot 2x\), which allows us to simplify the radical:
\[ -3\sqrt[3]{16x} = -3\sqrt[3]{8 \cdot 2x} = -3 \cdot \sqrt[3]{8} \cdot \sqrt[3]{2x} \]
Since \(\sqrt[3]{8} = 2\), the term becomes:
\[ -3 \cdot 2 \cdot \sqrt[3]{2x} = -6\sqrt[3]{2x} \]
For the third term, we observe that \(8x = 8 \cdot x\), so:
\[ -3\sqrt[3]{8x} = -3 \cdot \sqrt[3]{8} \cdot \sqrt[3]{x} \]
Since \(\sqrt[3]{8} = 2\), this term becomes:
\[ -3 \cdot 2 \cdot \sqrt[3]{x} = -6\sqrt[3]{x} \]
- Combining Like Terms and Final Simplification
Now, we substitute the simplified terms back into the original expression:
\[ 7\sqrt[3]{2x} - 6\sqrt[3]{2x} - 6\sqrt[3]{x} \]
We combine the terms involving \(\sqrt[3]{2x}\):
\[ (7 - 6)\sqrt[3]{2x} = 1\sqrt[3]{2x} = \sqrt[3]{2x} \]
The remaining term is \(-6\sqrt[3]{x}\). Combining these, the simplified form of the expression is:
\[ \sqrt[3]{2x} - 6\sqrt[3]{x} \]
Examples
Simplifying radical expressions is a common task in fields like engineering and physics, where complex formulas often need to be reduced to their simplest forms for easier calculation or better interpretation. For example, if you are calculating the volume of a sphere or the kinetic energy of an object where dimensions are expressed under roots, simplifying these expressions first can prevent errors and speed up the problem-solving process. Recognizing patterns, such as identifying perfect cubes within radical expressions, is a fundamental skill that makes these calculations more manageable.