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You uploaded a mathematical worksheet with the task of simplifying or rationalizing radicals. Let’s go through the steps for each of these items.

### 7a.
\[
\sqrt{15x^3y^3z} + 5 \sqrt{960xy^5z^3} + \sqrt{540x^5yz^5}
\]

To simplify this, we need to factor each radical and simplify the terms under the square roots.

### 7b.
\[
\sqrt[4]{15} \cdot \frac{\sqrt[4]{1875}}{\sqrt[4]{540}}
\]

We'll simplify the radicals by extracting factors that are perfect powers of 4.

### 7c.
\[
\sqrt[4]{3} \cdot \sqrt[5]{4}
\]

We'll combine these radicals using appropriate root laws.

### 7d.
\[
\frac{155 + \sqrt{7}}{\sqrt{7}}
\]

This can be simplified by rationalizing the denominator.

### 7e.
\[
\frac{\sqrt{7}}{155 + \sqrt{7}}
\]

Here, we rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator.

### 7f.
\[
\frac{\sqrt{x}}{27 + \sqrt{x}}
\]

Similarly, we rationalize the denominator here by multiplying both numerator and denominator by the conjugate.

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Here are 5 questions that could expand on the topic:
1. What is the process of rationalizing a denominator?
2. How do you handle cube roots versus square roots when simplifying radicals?
3. What is the conjugate of a binomial that includes a square root?
4. How do you simplify a fourth root, such as \(\sqrt[4]{1875}\)?
5. Why is it useful to simplify radicals when solving equations?

### Tip:
When simplifying radicals, always look for perfect squares or cubes (depending on the root) inside the radical that can be factored out.

Simplify or rationalize the following radicals as shown in the image.

This page provides step-by-step solutions for simplifying and rationalizing radicals from a Grade 10-12 math worksheet. The exercises involve using the laws of exponents, properties of radicals, and rationalization techniques for various expressions.

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