https://file.aiquickdraw.com/mathbot/file/1726193342213ks8s3mrs.jpg

The image shows a series of mathematical expressions involving square roots, where the task is to simplify or find the results of these operations. Let's go through each of them one by one to simplify the expressions:

### Given Expressions:
1. **a)** \((\sqrt{5} + \sqrt{7}) - (\sqrt{5} - \sqrt{7})\)
2. **b)** \((\sqrt{3} + \sqrt{5}) + (2\sqrt{3} + \sqrt{5})\)
3. **c)** \(3 - \sqrt{5} - (3 + \sqrt{5})\)
4. **d)** \(\sqrt{3} - \sqrt{7} + \sqrt{7} - \sqrt{5}\)
5. **e)** \((3 + \sqrt{5}) - (3 - \sqrt{5})\)
6. **f)** \(\frac{2\sqrt{3} + \sqrt{5}}{2\sqrt{3} + \sqrt{5}}\)

Let's solve each of these:

### 1. Expression (a):
\[
(\sqrt{5} + \sqrt{7}) - (\sqrt{5} - \sqrt{7}) 
= \sqrt{5} + \sqrt{7} - \sqrt{5} + \sqrt{7}
= 2\sqrt{7}
\]

### 2. Expression (b):
\[
(\sqrt{3} + \sqrt{5}) + (2\sqrt{3} + \sqrt{5}) 
= \sqrt{3} + 2\sqrt{3} + \sqrt{5} + \sqrt{5}
= 3\sqrt{3} + 2\sqrt{5}
\]

### 3. Expression (c):
\[
3 - \sqrt{5} - (3 + \sqrt{5}) 
= 3 - \sqrt{5} - 3 - \sqrt{5} 
= -2\sqrt{5}
\]

### 4. Expression (d):
\[
\sqrt{3} - \sqrt{7} + \sqrt{7} - \sqrt{5} 
= (\sqrt{3} - \sqrt{5})
\]

### 5. Expression (e):
\[
(3 + \sqrt{5}) - (3 - \sqrt{5}) 
= 3 + \sqrt{5} - 3 + \sqrt{5} 
= 2\sqrt{5}
\]

### 6. Expression (f):
\[
\frac{2\sqrt{3} + \sqrt{5}}{2\sqrt{3} + \sqrt{5}} = 1 \quad (\text{since any number divided by itself is 1, provided it's not zero})
\]

### Summary of Results:
1. **a)** \(2\sqrt{7}\)
2. **b)** \(3\sqrt{3} + 2\sqrt{5}\)
3. **c)** \(-2\sqrt{5}\)
4. **d)** \(\sqrt{3} - \sqrt{5}\)
5. **e)** \(2\sqrt{5}\)
6. **f)** \(1\)

Would you like further details on any of these steps, or do you have any questions?

### Follow-Up Questions:
1. How do you rationalize a denominator involving square roots?
2. Can you explain how to simplify expressions involving radicals?
3. What are the properties of square roots that are useful in simplifying expressions?
4. How can you determine if two radical expressions are equivalent?
5. Why does dividing a number by itself always result in 1 (assuming the number is non-zero)?

### Tip:
When simplifying expressions with square roots, combine like terms (terms with the same radical) to reduce the expression to its simplest form.

Simplify the following radical expressions: (\sqrt{5} + \sqrt{7}) - (\sqrt{5} - \sqrt{7}), (\sqrt{3} + \sqrt{5}) + (2\sqrt{3} + \sqrt{5}), 3 - \sqrt{5} - (3 + \sqrt{5}), \sqrt{3} - \sqrt{7} + \sqrt{7} - \sqrt{5}, (3 + \sqrt{5}) - (3 - \sqrt{5}), and \frac{2\sqrt{3} + \sqrt{5}}{2\sqrt{3} + \sqrt{5}}.

Learn how to simplify radical expressions like (\sqrt{5} + \sqrt{7}) - (\sqrt{5} - \sqrt{7}) and more. This guide covers key algebraic concepts, including properties of square roots, combining like terms, and rational expressions. Suitable for students in Grades 9-10.

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