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) $\quad 4 y \sqrt { 4 x y }$
B) $\quad 4 y \sqrt { 2 x y }$
C) $\quad 12 y \sqrt { 2 x y }$
D) $\quad 8 y \sqrt { 3 x y }$

Question

Quizplus · Accepted Answer

First, we simplify each term under the fourth root:
$$\sqrt [ 4 ] { 2 x y ^ { 5 } }=\sqrt { \sqrt { 2 x y ^ { 4 } } }=\sqrt {y \sqrt {2x}}$$
$$\sqrt [ 4 ] { 512 x y }=\sqrt { \sqrt { \sqrt { 2 ^ 9 } x y } } = 4\sqrt {2xy}$$
Note that we can simplify the first and last term as they are the same:
$$\sqrt {y\sqrt{2x}}-\sqrt {y\sqrt{2x}}=0$$
Now we substitute in the values we got:
\begin{align*}
\sqrt {y\sqrt{2x}}+4y\sqrt {2xy}&=\sqrt {2xy^{5/4}}+4y\sqrt{2xy}\&=y^{\frac{1}{4}}\sqrt {2x} \cdot \sqrt {y} + 4y\sqrt {2xy}\&=y^{1/4}\cdot\sqrt {2xy}(y^{1/4} + 4)\&=4y\sqrt {2xy}
\end{align*}
Therefore, the answer is (B).

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