- The expression \(\left(\frac{375}{343}\right)^{\frac{1}{3}}\) is equivalent to \(\sqrt[3]{\frac{375}{343}}\).
- Using radical properties, this simplifies to \(\frac{\sqrt[3]{375}}{\sqrt[3]{343}}\).
- Evaluating the cubes, \(\sqrt[3]{343} = 7\) and \(\sqrt[3]{375} = 5\sqrt[3]{3}\), so the expression is \(\frac{5}{7}\sqrt[3]{3}\).
- The correct choices are: \(\boxed{\frac{5}{7} \sqrt[3]{3}, \frac{\sqrt[3]{375}}{\sqrt[3]{343}}, \sqrt[3]{\frac{375}{343}}}\).
Explanation
- Simplifying the Expression using Radical Properties
We are given the expression \(\left(\frac{375}{343}\right)^{\frac{1}{3}}\). The exponent \(\frac{1}{3}\) corresponds to the cube root, so we can write the expression as \(\sqrt[3]{\frac{375}{343}}\). By the properties of radicals, specifically \(\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}\), we can rewrite this as \(\frac{\sqrt[3]{375}}{\sqrt[3]{343}}\).
- Evaluating the Cube Roots
Next, we evaluate the cube roots of the numerator and denominator. For the denominator, we recognize that \(343 = 7^3\), therefore \(\sqrt[3]{343} = 7\). For the numerator, we factor \(375\) as \(125 \times 3\). Since \(125 = 5^3\), we have \(\sqrt[3]{375} = \sqrt[3]{125 \times 3} = \sqrt[3]{125} \times \sqrt[3]{3} = 5\sqrt[3]{3}\). Substituting these values back into our fraction, we get \(\frac{5\sqrt[3]{3}}{7}\), which can also be written as \(\frac{5}{7} \sqrt[3]{3}\).
- Verifying the Given Options
We now evaluate the provided options:
- \(\frac{5}{7} \sqrt[3]{3}\): Matches our simplified expression.
- \(\frac{5}{7}\): Incorrect.
- \(\frac{375}{\sqrt[3]{343}} = \frac{375}{7}\): Incorrect.
- \(\frac{\sqrt[3]{375}}{\sqrt[3]{343}}\): This is equivalent to \(\sqrt[3]{\frac{375}{343}}\) by the property \(\frac{\sqrt[3]{a}}{\sqrt[3]{b}} = \sqrt[3]{\frac{a}{b}}\), which is equal to the original expression.
- \(\sqrt[3]{\frac{375}{343}}\): By definition, \(\left(\frac{375}{343}\right)^{\frac{1}{3}} = \sqrt[3]{\frac{375}{343}}\).
- \(\frac{\sqrt[3]{375}}{343} = \frac{5\sqrt[3]{3}}{343}\): Incorrect.
Therefore, the equivalent expressions are the first, fourth, and fifth options.
Examples
In engineering or scientific calculations, you often encounter complex ratios that need to be simplified to estimate values quickly. For example, if you are scaling a 3D model volume ratio \(\frac{375}{343}\), you might need to determine the ratio of the side lengths, which would involve the cube root \(\sqrt[3]{\frac{375}{343}}\). Knowing that this is equivalent to \(\frac{5}{7}\sqrt[3]{3}\) allows you to perform calculations with more manageable fractions and radicals.