https://r2.erweima.ai/mathbot/image/17242682492798dhmkjka.pdf

The document you uploaded is a math assignment focusing on the multiplication and division of radicals as well as the rationalization of denominators. Here's a breakdown of the problems provided:

### 1) Perform the following products and division of radicals:

a) 
\[
\frac{17 \sqrt{2}}{17 \sqrt{8}} \div \frac{\sqrt{5}}{\sqrt{5}}
\]

b)
\[
\frac{12 \sqrt{510} \times 7 \sqrt{2}}{4 \sqrt{9}} \div \frac{\sqrt{2}}{\sqrt{3}}
\]

c)
\[
\frac{12 \sqrt{515} \times 8 \sqrt{3}}{4 \sqrt{15}} \div \frac{3 \sqrt{9}}{\sqrt{2}}
\]

d)
\[
\frac{25 \sqrt{2} \times 15 \sqrt{8}}{54 \sqrt{518}}
\]

e)
\[
\frac{21 \sqrt{512} \times 514 \sqrt{5}}{6 \sqrt{5}} \div \frac{12 \sqrt{36}}{5 \sqrt{9}} \div \frac{\sqrt{2}}{\sqrt{5}}
\]

f)
\[
\frac{8 \sqrt{5} \times 28 \sqrt{514}}{5 \sqrt{3}} \div \frac{3 \sqrt{15}}{\sqrt{3}} \div \frac{\sqrt{9}}{\sqrt{5}}
\]

### 2) Rationalize the following denominators:

a) 
\[
\frac{\sqrt{2}}{5 \sqrt{2}}
\]

b) 
\[
\frac{5}{\sqrt{2} - \sqrt{2}}
\]

c) 
\[
\frac{\sqrt{169}}{\sqrt{213}}
\]

d)
\[
\frac{\sqrt{12}}{\sqrt{25} + \sqrt{2}}
\]

e)
\[
\frac{\sqrt{32}}{\sqrt{31} - \sqrt{2}}
\]

f)
\[
\frac{\sqrt{32}}{\sqrt{32} + \sqrt{2}}
\]

Would you like me to solve any specific problem or provide detailed solutions for these exercises? 

Here are some related questions you might find helpful:
1. What is the process of rationalizing a denominator?
2. How can you simplify the multiplication of two radicals?
3. What are the properties of radicals that apply during division?
4. Why is rationalization of a denominator important?
5. How can you identify like radicals before performing operations?

**Tip:** When rationalizing a denominator, multiplying both the numerator and the denominator by the conjugate of the denominator often simplifies the expression.

Explore multiplication and division of radicals with detailed solutions. Includes exercises on rationalizing denominators. Learn the process of simplifying radicals and why rationalization is important. Suitable for high school students.

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