https://file.aiquickdraw.com/mathbot/file/17259006899376xppa97f.jpg

Let's go through the steps to solve each part of the problem in the image:

### (a) Simplify:
\[
\sqrt{15x^3y^3z} + 5\sqrt{960xy^5z^3} + \sqrt{540x^5yz^5}
\]
We'll simplify each term inside the radicals:
1. \(\sqrt{15x^3y^3z} = \sqrt{15} \cdot x^{3/2} \cdot y^{3/2} \cdot \sqrt{z}\).
2. For \(5\sqrt{960xy^5z^3}\), simplify \(960 = 64 \times 15\), so:
   \[
   5\sqrt{960xy^5z^3} = 5\sqrt{64 \times 15 \cdot x \cdot y^5 \cdot z^3} = 5 \times 8 \times \sqrt{15} \cdot x^{1/2} \cdot y^{5/2} \cdot z^{3/2}.
   \]
3. \(\sqrt{540x^5yz^5} = \sqrt{540} \cdot x^{5/2} \cdot y^{1/2} \cdot z^{5/2}\), and \(540 = 36 \times 15\), so:
   \[
   \sqrt{540x^5yz^5} = 6\sqrt{15} \cdot x^{5/2} \cdot y^{1/2} \cdot z^{5/2}.
   \]

Collect like terms with \(\sqrt{15}\) and combine powers of variables.

### (b) Simplify:
\[
\sqrt[4]{15} \cdot \sqrt[4]{\frac{1875}{540}}
\]
Break down into:
- \(\sqrt[4]{15}\) remains as is.
- Simplify \(\frac{1875}{540} = \frac{375}{108} = \frac{125}{36}\), so:
  \[
  \sqrt[4]{\frac{1875}{540}} = \frac{\sqrt[4]{125}}{\sqrt[4]{36}} = \frac{5^{3/4}}{6^{1/2}}.
  \]
Final expression: \(\sqrt[4]{15} \cdot \frac{5^{3/4}}{6^{1/2}}\).

### (c) Simplify:
\[
\sqrt[4]{3} \cdot \sqrt[5]{4}
\]
Write as:
\[
(3^{1/4}) \cdot (4^{1/5}).
\]
This cannot be further simplified and remains in the product of radicals.

### (d) Rationalize:
\[
\frac{155 + \sqrt{7}}{\sqrt{7}}
\]
Multiply numerator and denominator by \(\sqrt{7}\):
\[
\frac{(155 + \sqrt{7})\sqrt{7}}{7} = \frac{155\sqrt{7} + 7}{7}.
\]

### (e) Rationalize:
\[
\frac{\sqrt{7}}{155 + \sqrt{7}}
\]
Multiply by the conjugate \(155 - \sqrt{7}\):
\[
\frac{\sqrt{7}(155 - \sqrt{7})}{(155 + \sqrt{7})(155 - \sqrt{7})} = \frac{155\sqrt{7} - 7}{155^2 - 7}.
\]

### (f) Rationalize:
\[
\frac{\sqrt{x}}{27 + \sqrt{x}}
\]
Multiply by the conjugate \(27 - \sqrt{x}\):
\[
\frac{\sqrt{x}(27 - \sqrt{x})}{(27 + \sqrt{x})(27 - \sqrt{x})} = \frac{27\sqrt{x} - x}{729 - x}.
\]

Would you like further details on any specific part?

Here are 5 relative questions:
1. Can all radicals in part (a) be combined further?
2. What is the rule for rationalizing the denominator?
3. How do fractional exponents help simplify radicals?
4. Why can't we simplify part (c) further?
5. Can rationalization always be used to simplify expressions?

**Tip:** For complex radicals, always look for factorization to simplify easily!

Simplify or rationalize the following radicals.

Learn how to simplify and rationalize radicals with detailed steps for each problem. This lesson covers simplifying square roots, multiplying radical expressions, and rationalizing denominators. Perfect for students in Grades 9-12.

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