If $x
eq 0$, what is the sum of $4 \sqrt[3]{x^{10}}+5 x^3 \sqrt[3]{8 x}$ in simplest form?
To simplify the given expression \(4 \sqrt[3]{x^{10}} + 5x^3 \sqrt[3]{8x}\) where \(x eq 0\), we begin by examining each term separately. We will use the properties of radicals and exponents, specifically that \(\sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b}\) and \(\sqrt[n]{a^n} = a\).
Consider the first term: \(4 \sqrt[3]{x^{10}}\). We can rewrite \(x^{10}\) as \(x^9 \cdot x\). Since \(x^9 = (x^3)^3\), the cube root is \(x^3\). Thus,
\[\sqrt[3]{x^{10}} = \sqrt[3]{x^9 \cdot x} = \sqrt[3]{x^9} \cdot \sqrt[3]{x} = x^3 \sqrt[3]{x}.\]
Multiplying this by the coefficient 4, we get:
\[4 \sqrt[3]{x^{10}} = 4x^3 \sqrt[3]{x}.\]
Consider the second term: \(5x^3 \sqrt[3]{8x}\). We know that \(\sqrt[3]{8} = 2\). Applying the radical property, we have:
\[\sqrt[3]{8x} = \sqrt[3]{8} \cdot \sqrt[3]{x} = 2 \sqrt[3]{x}.\]
Substituting this back into the second term:
\[5x^3 \sqrt[3]{8x} = 5x^3 \cdot (2 \sqrt[3]{x}) = 10x^3 \sqrt[3]{x}.\]
Finally, we combine the two simplified terms:
\[4x^3 \sqrt[3]{x} + 10x^3 \sqrt[3]{x}.\]
Since both terms contain the common factor \(x^3 \sqrt[3]{x}\), we can add their coefficients:
\[(4 + 10) x^3 \sqrt[3]{x} = 14x^3 \sqrt[3]{x}.\]
Thus, the expression in simplest form is \(\boxed{14x^3 \sqrt[3]{x}}\).
In many engineering and physics applications, you often encounter complex expressions involving radicals that represent physical dimensions or energy states. Simplifying these expressions is crucial for performing calculations efficiently and identifying relationships between different variables. For example, if an expression for volume is given by a formula involving cube roots of variable side lengths, simplifying it allows for easier calculation of the total volume as the variables change over time. Being able to combine like terms, as done here, is a fundamental skill for such tasks.