https://file.aiquickdraw.com/mathbot/file/1727589247766m03eouyi.jpg

Let's go ahead and simplify the given square root expressions step by step.

### (a) \( 4\sqrt{48} + 3\sqrt{147} - 5\sqrt{12} \)

We first simplify each square root:
- \( \sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3} \)
- \( \sqrt{147} = \sqrt{49 \times 3} = 7\sqrt{3} \)
- \( \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3} \)

Now substitute these into the expression:
\[
4(4\sqrt{3}) + 3(7\sqrt{3}) - 5(2\sqrt{3}) = 16\sqrt{3} + 21\sqrt{3} - 10\sqrt{3}
\]
Simplifying the terms:
\[
16\sqrt{3} + 21\sqrt{3} - 10\sqrt{3} = 27\sqrt{3}
\]

### (b) \( \sqrt{2} - 8\sqrt{2} \)

Combine the like terms:
\[
\sqrt{2} - 8\sqrt{2} = -7\sqrt{2}
\]

### (c) \( \sqrt{8} - 3\sqrt{2} \)

First, simplify \( \sqrt{8} \):
\[
\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}
\]
Now substitute and combine the terms:
\[
2\sqrt{2} - 3\sqrt{2} = -\sqrt{2}
\]

### (d) \( \sqrt{50} - \sqrt{32} \)

Simplify each square root:
- \( \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2} \)
- \( \sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2} \)

Now subtract:
\[
5\sqrt{2} - 4\sqrt{2} = \sqrt{2}
\]

### (e) \( \sqrt{90} - 5\sqrt{40} - 2\sqrt{10} \)

Simplify each square root:
- \( \sqrt{90} = \sqrt{9 \times 10} = 3\sqrt{10} \)
- \( \sqrt{40} = \sqrt{4 \times 10} = 2\sqrt{10} \)

Substitute into the expression:
\[
3\sqrt{10} - 5(2\sqrt{10}) - 2\sqrt{10} = 3\sqrt{10} - 10\sqrt{10} - 2\sqrt{10}
\]
Combine like terms:
\[
3\sqrt{10} - 10\sqrt{10} - 2\sqrt{10} = -9\sqrt{10}
\]

### (f) \( \sqrt{2} - 2\sqrt{2} - 3\sqrt{8} \)

First simplify \( \sqrt{8} \):
\[
\sqrt{8} = 2\sqrt{2}
\]
Now substitute and combine:
\[
\sqrt{2} - 2\sqrt{2} - 3(2\sqrt{2}) = \sqrt{2} - 2\sqrt{2} - 6\sqrt{2}
\]
\[
\sqrt{2} - 2\sqrt{2} - 6\sqrt{2} = -7\sqrt{2}
\]

### (g) \( 2\sqrt{5} + \sqrt{45} \)

Simplify \( \sqrt{45} \):
\[
\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}
\]
Now substitute and combine the terms:
\[
2\sqrt{5} + 3\sqrt{5} = 5\sqrt{5}
\]

### (h) \( \sqrt{5} + 4\sqrt{5} + 3\sqrt{125} \)

Simplify \( \sqrt{125} \):
\[
\sqrt{125} = \sqrt{25 \times 5} = 5\sqrt{5}
\]
Now substitute and combine:
\[
\sqrt{5} + 4\sqrt{5} + 3(5\sqrt{5}) = \sqrt{5} + 4\sqrt{5} + 15\sqrt{5}
\]
\[
\sqrt{5} + 4\sqrt{5} + 15\sqrt{5} = 20\sqrt{5}
\]

### (i) \( \sqrt{12} - \sqrt{27} \)

Simplify both square roots:
- \( \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3} \)
- \( \sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3} \)

Now subtract:
\[
2\sqrt{3} - 3\sqrt{3} = -\sqrt{3}
\]

### (j) \( 2\sqrt{3} - \sqrt{48} \)

Simplify \( \sqrt{48} \):
\[
\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}
\]
Now substitute and subtract:
\[
2\sqrt{3} - 4\sqrt{3} = -2\sqrt{3}
\]

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Do you have any questions or would you like further clarification?

Here are 5 related questions you might explore:
1. How do you simplify square roots with non-perfect square factors?
2. What is the process for combining like terms with square roots?
3. How can you simplify expressions with cube roots or higher-order roots?
4. What are the key properties of radicals that help in simplification?
5. How do you rationalize the denominator when dealing with square roots?

**Tip:** Always try to break square roots into their prime factorization to simplify them easily!

Simplify the given square root expressions, including terms such as 4√48 + 3√147 - 5√12 and others.

Learn how to simplify square root expressions such as 4√48 + 3√147 - 5√12 and others with detailed step-by-step solutions. This problem involves simplifying radicals, breaking down square roots, and combining like terms. Suitable for high school students.

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