https://file.aiquickdraw.com/mathbot/file/1728600466696i38pttbx.jpg

The task is to simplify the given radicals by extracting the factors outside the root.

Here are the problems from the image and their solutions:

### 1) \(\sqrt[4]{80}\)
We can start by factoring 80 into prime factors:  
\(80 = 2^4 \cdot 5\).

Now, apply the fourth root:
\[
\sqrt[4]{80} = \sqrt[4]{2^4 \cdot 5} = 2 \cdot \sqrt[4]{5}.
\]
So, the simplified form is:
\[
2\sqrt[4]{5}.
\]

### 2) \(\sqrt[3]{432}\)
First, factor 432 into prime factors:  
\(432 = 2^4 \cdot 3^3\).

Now, apply the cube root:
\[
\sqrt[3]{432} = \sqrt[3]{2^4 \cdot 3^3} = 3 \cdot \sqrt[3]{2^4} = 3 \cdot 2 \cdot \sqrt[3]{2} = 6\sqrt[3]{2}.
\]
So, the simplified form is:
\[
6\sqrt[3]{2}.
\]

### 3) \(\sqrt[3]{54y^8}\)
First, factor 54:  
\(54 = 2 \cdot 3^3\).

Also, split \(y^8\) as \(y^8 = (y^6) \cdot y^2\).

Now, apply the cube root:
\[
\sqrt[3]{54y^8} = \sqrt[3]{2 \cdot 3^3 \cdot y^6 \cdot y^2} = 3y^2 \cdot \sqrt[3]{2y^2}.
\]
So, the simplified form is:
\[
3y^2\sqrt[3]{2y^2}.
\]

---

Would you like further details on any of these steps or concepts?

Here are five related questions you might find interesting:
1. How do you simplify higher-order roots like \(\sqrt[5]{x}\)?
2. How can you factor expressions under a square root to simplify them?
3. What are the properties of cube roots and how are they applied?
4. How can we work with variables in radical expressions?
5. What is the difference between simplifying square roots and cube roots?

**Tip**: Always check for perfect powers (like \(2^3, 3^3\)) under the root when simplifying radicals.

Винесіть множник з-під знака кореня: 1) √[4]{80}, 2) √[3]{432}, 3) √[3]{54y^8}

Learn how to simplify the radicals √[4]{80}, √[3]{432}, and √[3]{54y^8} by factoring and applying the properties of roots. This problem involves prime factorization and is suitable for students in Grades 9-11.

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