Problem 74 - College Algebra

Question

Simplify the radical expressions if possible. $$\frac{\sqrt[4]{162 x^{5}}}{\sqrt[4]{2 x}}$$

Step-by-Step Solution

Answer

The simplified form of the expression is \(3x\).

Step 1: Simplify Numerical Part of the Radical

We first deal with the numeric part of the expression under the fourth root. In the numerator, divide 162 by 2, getting 81. The expression becomes: \(\frac{\sqrt[4]{81x^{5}}}{\sqrt[4]{x}}\).

Step 2: Simplify Algebric Part of the Radical

Next, we simplify the algebraic part of the expression under the fourth root. Since \(x^{5}\) is divided by \(x\), we adjust the powers and get \(x^{4}\). So, the expression now becomes \(\frac{\sqrt[4]{81x^{4}}}{1}\).

Step 3: Simplify Numerical Radical

We can simplify \(\sqrt[4]{81}\) to be 3, since 3^4 equals 81. The expression is now \(\frac{3x}{1}\).

Step 4: Final Simplification

Given that any number divided by 1 remains the same number, we can simplify our expression to 3x.