Simplify the expression.Assume that all variables represent positive numbers, so that no absolute value symbols are needed. $$\sqrt [ 4 ] { 2 x y ^ { 5 } } + y \sqrt [ 4 ] { 512 x y } - \sqrt [ 4 ] { 2 x y ^ { 5 } }$$
A) $\quad 8 y \sqrt { 3 x y }$
B) $\quad 12 y \sqrt { 2 x y }$
C) $\quad 4 y \sqrt { 4 x y }$
D) $\quad 4 y \sqrt { 2 x y }$

Question

Quizplus · Accepted Answer

We can simplify each term under the fourth root sign first.
$$\sqrt [ 4 ] { 2 x y ^ { 5 } } = \sqrt { \sqrt [ 4 ] { 2 x y ^ { 5 } } \cdot \sqrt [ 4 ] { 2 x y ^ { 5 } } } = \sqrt { \sqrt [ 4 ] { (2xy^5)^2 } } = \sqrt { 2xy^2 }$$
$$\sqrt [ 4 ] { 512 x y } = \sqrt { \sqrt { 512xy } }$$
Since $512=2^9$, we can simplify this as follows:
$$\sqrt [ 4 ] { 512 x y } = \sqrt { \sqrt { 2^9xy } } = \sqrt { \sqrt { 512xy } } = \sqrt { 16xy } = 4\sqrt{xy}$$
Now we can substitute these simplifications into our original expression:
$$\sqrt { 2xy^2 } + y\cdot 4\sqrt { xy } - \sqrt { 2xy^2 } = 4y\sqrt { xy }$$

Simplify the Expression 2xy54+y512xy4−2xy54\sqrt [ 4 ] { 2 x y ^ { 5 } } + y \sqrt [ 4 ] { 512 x y } - \sqrt [ 4 ] { 2 x y ^ { 5 } }42xy5​+y4512xy​−42xy5​

Simplify the Expression $\sqrt [ 4 ] { 2 x y ^ { 5 } } + y \sqrt [ 4 ] { 512 x y } - \sqrt [ 4 ] { 2 x y ^ { 5 } }$