Simplify the following radicals.

a. radical over 40 = 
 
  

b. radical over 148=
 
  


To simplify the given radicals, let's break down each number into its prime factors and then simplify.

### a. \(\sqrt{40}\)
First, factorize 40:
\[
40 = 2 \times 2 \times 2 \times 5 = 2^3 \times 5
\]
Now, take the square root of 40:
\[
\sqrt{40} = \sqrt{2^3 \times 5} = \sqrt{2^2 \times 2 \times 5} = \sqrt{2^2} \times \sqrt{2 \times 5} = 2\sqrt{10}
\]
So, \(\sqrt{40} = 2\sqrt{10}\).

### b. \(\sqrt{148}\)
First, factorize 148:
\[
148 = 2 \times 2 \times 37 = 2^2 \times 37
\]
Now, take the square root of 148:
\[
\sqrt{148} = \sqrt{2^2 \times 37} = \sqrt{2^2} \times \sqrt{37} = 2\sqrt{37}
\]
So, \(\sqrt{148} = 2\sqrt{37}\).

### Final Answers:
- a. \(\sqrt{40} = 2\sqrt{10}\)
- b. \(\sqrt{148} = 2\sqrt{37}\)

Would you like more details on any of these steps or have any questions?

Here are 5 related questions you might find useful:
1. How do you simplify \(\sqrt{72}\)?
2. What is the simplified form of \(\sqrt{50}\)?
3. How do you factorize the number 180 to simplify \(\sqrt{180}\)?
4. Can you simplify the expression \(\sqrt{200}\)?
5. How would you simplify the radical \(\sqrt{18}\)?

**Tip:** When simplifying radicals, always look for the largest perfect square factor of the number under the radical sign.

Learn how to simplify radicals √40 and √148 with detailed step-by-step solutions. This problem involves prime factorization and square roots. Suitable for students in Grades 8-10.

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