What is the following product?
$\sqrt[3]{4} \cdot \sqrt{3}$
To solve the expression \(\sqrt[3]{4} \cdot \sqrt{3}\), we first express both radicals with fractional exponents to identify a common root index.
Recall that \(\sqrt[n]{x} = x^{1/n}\).
Thus, \(\sqrt[3]{4} = 4^{1/3}\) and \(\sqrt{3} = 3^{1/2}\).
To multiply these terms, we need a common denominator for the exponents, which is the least common multiple of 3 and 2, which is 6.
Rewriting the exponents:
\(4^{1/3} = 4^{2/6} = (4^2)^{1/6} = 16^{1/6} = \sqrt[6]{16}\)
\(3^{1/2} = 3^{3/6} = (3^3)^{1/6} = 27^{1/6} = \sqrt[6]{27}\)
Now the product is \(\sqrt[6]{16} \cdot \sqrt[6]{27}\).
Using the property of radicals \(\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}\), we have:
\(\sqrt[6]{16} \cdot \sqrt[6]{27} = \sqrt[6]{16 \cdot 27} = \sqrt[6]{432}\).
Thus, the product is \(\sqrt[6]{432}\).
Understanding how to work with radicals of different orders is essential in fields like engineering and physics, where complex formulas involve roots. For example, when calculating the resonant frequency of an oscillator or the effective capacity in a series-parallel circuit, you might encounter roots of different degrees. Converting them to a common radical index allows you to combine and simplify these expressions, making it easier to interpret the final physical values and design systems more efficiently.