Question

What is the simplified form of the following expression? Assume x greater-than-or-equal-to 0 and y greater-than-or-equal-to 0

2 (RootIndex 4 StartRoot 16 x EndRoot) minus 2 (RootIndex 4 StartRoot 2 y EndRoot) + 3 (RootIndex 4 StartRoot 81 x EndRoot) minus 4 (RootIndex 4 StartRoot 32 y EndRoot)
5 (RootIndex 4 StartRoot x EndRoot) minus 4 (RootIndex 4 StartRoot 32 y EndRoot)
5 (RootIndex 4 StartRoot x EndRoot) minus 6 (RootIndex 4 StartRoot 2 y EndRoot)
13 (RootIndex 4 StartRoot x EndRoot) minus 10 (RootIndex 4 StartRoot 2 y EndRoot)
35 (RootIndex 4 StartRoot x EndRoot) minus 18 (RootIndex 4 StartRoot 2 y EndRoot)

Answer 1

`\text{(c) }\ 13\sqrt[4]{x}-10\sqrt[4]{2y}`

Explanation:

Simplification of radical expressions of this sort involves ...

• factoring out integer powers
• combining like terms

The given expression can be simplified as follows:

`\displaystyle 2\sqrt[4]{16x}-2\sqrt[4]{2y}+3\sqrt[4]{81x}-4\sqrt[4]{32y}\\\\=2\sqrt[4]{2^4x}-2\sqrt[4]{2y}+3\sqrt[4]{3^4x}-4\sqrt[4]{2^4(2y)}\\\\=2\cdot2\sqrt[4]{x}-2\sqrt[4]{2y}+3\cdot3\sqrt[4]{x}-4\cdot2\sqrt[4]{2y}\\\\=(2\cdot2+3\cdot3)\sqrt[4]{x}-(2+4\cdot2)\sqrt[4]{2y}\\\\=\boxed{13\sqrt[4]{x}-10\sqrt[4]{2y}}`