\(\sqrt[4]{256x^{3}y^{12}}\)

Answer 1

To simplify \(\sqrt[4]{256x^{3}y^{12}}\), we'll break it down into components.

1. Simplify the constant term: \[ 256 = 2^8 \quad \text{(since \(2^8 = 256\))} \] Therefore: \[ \sqrt[4]{256} = \sqrt[4]{2^8} = 2^{8/4} = 2^2 = 4. \]

2. Simplify the \(x^3\) term: \[ \sqrt[4]{x^3} = x^{3/4}. \]

3. Simplify the \(y^{12}\) term: \[ \sqrt[4]{y^{12}} = y^{12/4} = y^3. \]

Now, combining all parts, we have: \[ \sqrt[4]{256x^{3}y^{12}} = \sqrt[4]{256} \cdot \sqrt[4]{x^{3}} \cdot \sqrt[4]{y^{12}} = 4 \cdot x^{3/4} \cdot y^{3}. \]

Thus, the simplified form is: \[ \sqrt[4]{256x^{3}y^{12}} = 4y^3x^{3/4}. \]