Simplify the radical.

Question

16xy6−−−−−√3
16
𝑥
𝑦
6
3
(1 point)
Responses
4y2x−−√3
4
𝑦
2
𝑥
3
4 y squared cube root of x
2xy22–√3
2
𝑥
𝑦
2
2
3
2 x y squared cube root of 2
2y22x−−√3
2
𝑦
2
2
𝑥
3
2 y squared cube root of 2 x
8y22x−−√3

Answer 1

To simplify the expression \( \sqrt[3]{16xy^6} \), we can break it down into its prime factors and simplify accordingly.

First, let's rewrite \( 16 \) and \( y^6 \):

1. \( 16 = 2^4 \)
2. \( y^6 = (y^2)^3 \)

Now, we can rewrite the expression:

\[ \sqrt[3]{16xy^6} = \sqrt[3]{2^4 \cdot x \cdot (y^2)^3} \]

Applying the property of radicals, we can separate this into two parts:

\[ \sqrt[3]{2^4} \cdot \sqrt[3]{x} \cdot \sqrt[3]{(y^2)^3} \]

We know that \( \sqrt[3]{(y^2)^3} = y^2 \). Now for \( \sqrt[3]{2^4} \), we can simplify it:

\[ 2^4 = 2^3 \cdot 2^1 \implies \sqrt[3]{2^4} = 2 \cdot \sqrt[3]{2} \]

Putting it all together, we have:

\[ \sqrt[3]{16xy^6} = 2 \cdot \sqrt[3]{2} \cdot y^2 \cdot \sqrt[3]{x} = 2y^2 \sqrt[3]{2x} \]

That means the simplified form of the original expression is:

\[ 2y^2 \sqrt[3]{2x} \]

Since none of the given options directly matches with \( 2y^2 \sqrt[3]{2x} \), it's possible that there may have been a typo or misunderstanding in the options provided.

Please double-check the options to ensure there are no mistakes, or let me know if there are any clarifications on the format of the result you’re looking for.

Answer 2

Simplify the radical expression.
−2–√+8–√−320−−√
−
2
+
8
−
3
20
(1 point)
Responses
−53–√
−
5
3
negative 5 square root of 3
32–√−65–√
3
2
−
6
5
3 square root of 2 minus 6 square root of 5
4–√−65–√
4
−
6
5
square root of 4 minus 6 square root of 5
2–√−65–√

Answer 3

To simplify the expression \(-\sqrt{2} + \sqrt{8} - \sqrt{320}\), we need to simplify the square roots involved.

1. Simplify \(\sqrt{8}\): \[ \sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2} \]

2. Simplify \(\sqrt{320}\): \[ \sqrt{320} = \sqrt{64 \cdot 5} = \sqrt{64} \cdot \sqrt{5} = 8\sqrt{5} \]

Now we can substitute these simplified radicals back into the expression: \[ -\sqrt{2} + 2\sqrt{2} - 8\sqrt{5} \]

Next, combine the like terms: \[ (-\sqrt{2} + 2\sqrt{2}) = (2 - 1)\sqrt{2} = \sqrt{2} \]

Thus, the expression simplifies to: \[ \sqrt{2} - 8\sqrt{5} \]

Now, let's look at your response options. Unfortunately, none of the given options exactly match \(\sqrt{2} - 8\sqrt{5}\). Let’s verify the structure of our expression:

The final simplified result is:

\(\sqrt{2} - 8\sqrt{5}\)

If any adjustments or confirmations are needed on the options provided, please let me know!