91Ó°ÊÓ

Function multiplication involves multiplying two functions together to create a new function. In our exercise, we have two functions: \( f(x) = \sqrt[4]{2x} + 5\sqrt{2x} \) and \( g(x) = \sqrt[3]{2x} \). To find \( (f \cdot g)(x) \), we need to multiply these functions: \( f(x) \cdot g(x) \). This means we substitute the expressions given for each function and then multiply them: \( (\sqrt[4]{2x} + 5\sqrt{2x}) \cdot \sqrt[3]{2x} \). Distributing the product, each term in the sum \( \sqrt[4]{2x} + 5\sqrt{2x} \) is multiplied by \( \sqrt[3]{2x} \):\((\sqrt[4]{2x} \cdot \sqrt[3]{2x}) + (5\sqrt{2x} \cdot \sqrt[3]{2x}) \).

When working with exponents, knowing the properties can make simplifying expressions easier. One key property is \( a^{m/n} \cdot a^{p/q} = a^{(m/n + p/q)} \). For the term \( \sqrt[4]{2x} \cdot \sqrt[3]{2x} \), we rewrite each radical as an exponent and then add those exponents: \( (2x)^{1/4} \cdot (2x)^{1/3} = (2x)^{1/4 + 1/3} \). Use a common denominator to add \( \frac{1}{4} + \frac{1}{3} = \frac{3}{12} + \frac{4}{12} = \frac{7}{12} \), thus \( (2x)^{7/12} \). For the term \( 5\sqrt{2x} \cdot \sqrt[3]{2x} \), rewrite as \( 5(2x)^{1/2} \cdot (2x)^{1/3} \) and add \( \frac{1}{2} + \frac{1}{3} = \frac{3/6} + \frac{2/6} = \frac{5/6} \), resulting in \( 5(2x)^{5/6} \). Remember these exponent properties are very useful when dealing with radicals.

Radicals are expressions that involve roots, such as square roots \( \sqrt{} \), cube roots \( \sqrt[3]{} \), and fourth roots \( \sqrt[4]{} \). To simplify products involving radicals, convert them to rational exponents. For example, \( \sqrt[4]{2x} \) can be written as \( (2x)^{1/4} \). Similarly, \( \sqrt[3]{2x} \) is \( (2x)^{1/3} \). When multiplying these, use the exponent properties discussed earlier. For the problem in the exercise, multiply \( \sqrt[4]{2x} + 5\sqrt{2x} \) with \( \sqrt[3]{2x} \) and simplify each term: \( \sqrt[4]{2x} \cdot \sqrt[3]{2x} \) becomes \( (2x)^{7/12} \) and \( 5\sqrt{2x} \cdot \sqrt[3]{2x} = 5(2x)^{5/6} \). Combining these gives \( (2x)^{7/12} + 5(2x)^{5/6} \). Understanding radicals as exponents helps in simplifying and performing operations on these expressions.