https://file.aiquickdraw.com/mathbot/file/1727057274141496179sp.jpg

You’ve provided an image containing five expressions involving radicals and fractions. Let’s simplify each of them:

### a) \(\frac{1}{\sqrt{6}}\)

This is already simplified. Another way to express this could be:
\[
\frac{1}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}} = \frac{\sqrt{6}}{6}
\]
So, the simplified form is \(\frac{\sqrt{6}}{6}\).

---

### b) \(\frac{5}{3\sqrt{x}}\)

We can rationalize the denominator by multiplying both the numerator and denominator by \(\sqrt{x}\):
\[
\frac{5}{3\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}} = \frac{5\sqrt{x}}{3x}
\]
Thus, the simplified form is \(\frac{5\sqrt{x}}{3x}\).

---

### c) \(\frac{8}{\sqrt{2y}}\)

Similar to part (b), we will rationalize the denominator by multiplying by \(\sqrt{2y}\):
\[
\frac{8}{\sqrt{2y}} \cdot \frac{\sqrt{2y}}{\sqrt{2y}} = \frac{8\sqrt{2y}}{2y} = \frac{4\sqrt{2y}}{y}
\]
So, the simplified form is \(\frac{4\sqrt{2y}}{y}\).

---

### d) \(\frac{2}{3\sqrt[3]{2}}\)

This involves a cube root in the denominator. It’s already quite simplified, but we can express the denominator as \(3 \cdot 2^{1/3}\). There is no further simplification needed unless you'd like to convert it to decimal form or approximate \(2^{1/3}\).

---

### e) \(\sqrt[3]{\frac{32}{x}}\)

The cube root applies to both the numerator and the denominator:
\[
\sqrt[3]{\frac{32}{x}} = \frac{\sqrt[3]{32}}{\sqrt[3]{x}} = \frac{2\sqrt[3]{4}}{\sqrt[3]{x}}
\]
Thus, the simplified form is \(\frac{2\sqrt[3]{4}}{\sqrt[3]{x}}\).

---

Let me know if you would like further details on any of these simplifications or if you have any questions.

### 5 Related Questions:
1. How do you rationalize the denominator for expressions involving cube roots?
2. Can you simplify \(\frac{7}{\sqrt{5x}}\)?
3. What happens when you multiply two square root expressions together?
4. How do you simplify expressions involving higher-order roots (e.g., fourth roots)?
5. What is the best way to handle cube roots when simplifying fractions?

### Tip:
Always remember to rationalize the denominator by multiplying by an appropriate factor to make the denominator a rational number or higher-order expression.

Simplify the given radical and fraction expressions: 1/sqrt(6), 5/(3sqrt(x)), 8/sqrt(2y), 2/(3cbrt(2)), and cbrt(32/x).

Learn how to simplify expressions involving square and cube roots, such as 1/sqrt(6), 5/(3sqrt(x)), and cbrt(32/x). The solution explains rationalizing denominators and simplifying radicals step-by-step for students in Grades 7-9.

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