Questions: Simplify the radical expression. (Use absolute value signs, if appropriate.) √[4]5 x^4 y^6

Solution Steps

To simplify the given radical expression, we need to break down the expression inside the fourth root and simplify each component separately. We will use the properties of exponents and radicals to achieve this.

1. Recognize that \(\sqrt[4]{a^b} = a^{b/4}\).
2. Apply this property to each term inside the radical.
3. Simplify the resulting expression, using absolute value signs where necessary.

We start with the expression: \[ \sqrt[4]{5 x^{4} y^{6}} = \sqrt[4]{5} \cdot \sqrt[4]{x^{4}} \cdot \sqrt[4]{y^{6}} \]

Using the property \(\sqrt[4]{a^b} = a^{b/4}\), we simplify each term: \[ \sqrt[4]{x^{4}} = x^{4/4} = x \] \[ \sqrt[4]{y^{6}} = y^{6/4} = y^{3/2} \]

Now we can combine the simplified components: \[ \sqrt[4]{5 x^{4} y^{6}} = \sqrt[4]{5} \cdot x \cdot y^{3/2} \]

Final Answer