Simplify.
$-\sqrt[3]{40}+3 \sqrt[3]{320}$
The objective is to simplify the expression \(-\sqrt[3]{40} + 3\sqrt[3]{320}\). To do this, we need to extract perfect cube factors from the radicands (the numbers under the cube roots) 40 and 320. A perfect cube is a number of the form \(k^3\), such as \(1, 8, 27, 64, 125, \dots\). We will use the property of radicals \(\sqrt[n]{x \cdot y} = \sqrt[n]{x} \cdot \sqrt[n]{y}\) to simplify each term.
First, we factor 40 to identify any perfect cube factors:
\(40 = 8 \times 5 = 2^3 \times 5\).
Now, apply the radical property:
\(\sqrt[3]{40} = \sqrt[3]{8 \times 5} = \sqrt[3]{8} \cdot \sqrt[3]{5} = 2\sqrt[3]{5}\).
Next, we factor 320 to identify any perfect cube factors:
\(320 = 64 \times 5 = 4^3 \times 5\).
Now, apply the radical property:
\(\sqrt[3]{320} = \sqrt[3]{64 \times 5} = \sqrt[3]{64} \cdot \sqrt[3]{5} = 4\sqrt[3]{5}\).
Substitute the simplified radicals back into the original expression:
\(-\sqrt[3]{40} + 3\sqrt[3]{320} = -(2\sqrt[3]{5}) + 3(4\sqrt[3]{5})\).
Multiply the constants:
\(= -2\sqrt[3]{5} + 12\sqrt[3]{5}\).
Combine the like terms by adding their coefficients:
\(= (-2 + 12)\sqrt[3]{5} = 10\sqrt[3]{5}\).
The simplified expression is \(10\sqrt[3]{5}\).
In many fields like engineering or physics, simplified radicals are essential for precise calculations. For example, if you are calculating the volume of a complex geometric shape where side lengths are given as cube roots, simplifying those roots into a common form allows for easier addition and comparison of dimensions. This algebraic skill ensures that complex expressions remain manageable and accurate for further analysis.