Please help me simplify these following mixed radicals:: and show me how you did it cause I need to learn :

I don't know how to type a square root sign so I just wrote square root and the numbers infront of the square root are supposed to be multiplied.
2 square root 48
3 square root 81
6 square root 12
3 square root 32
2 square root 18
5 square root 48
3 square root 54
Thanks :)

To do this, you have to factor the radicand (or the number inside the radical sign) and look for perfect squares. For instance,

2 * √(48)
2 * √(16*3)
2 * √(4*4*3)
Express the repeating factors using exponents, so it's easier to see. Since four is multiplied by itself twice,
2 * √((4^2) * 3)
The 4^2 is a perfect square, it's squareroot is equal to 4. Therefore you have,
2 * 4 √(3)
= 8 * √(3)

#2.
3 * √(81)
3 * √(9*9)
3 * √(9^2)
3 * 9
= 27

#3.
6 * √(12)
6 * √(2*2*3)
6 * √((2^2) * 3)
6 * 2 * √(3)
= 12 * √(3)

#4.
3 * √(32)
3 * √(8*4)
3 * √(2*4*4)
3 * √((4^2) * 2)
3 * 4 * √(2)
= 12 * √(2)

Now, try solving the rest.
Hope this helps~ :3

the square root of 20 minus the sqare root of 5 plus the square root of 45

To simplify mixed radicals, we want to factor the numbers inside the square root symbols and find the perfect square factors. Let's simplify each of the given mixed radicals one by one:

1. 2√48:
- We start by factoring 48: 48 = 2 * 24 = 2 * 2 * 12 = 2 * 2 * 2 * 6.
- Now we can rewrite the mixed radical as follows: 2√48 = 2√(2 * 2 * 2 * 6) = 2 * 2√6 = 4√6.

2. 3√81:
- The number 81 is already a perfect square, as 9 * 9 = 81.
- Hence, 3√81 = 3 * 9 = 27.

3. 6√12:
- Similar to the first example, we need to factor 12: 12 = 2 * 6 = 2 * 2 * 3.
- Rewriting the mixed radical gives us: 6√12 = 6√(2 * 2 * 3) = 6 * 2√3 = 12√3.

4. 3√32:
- Now, let's factor 32: 32 = 2 * 16 = 2 * 2 * 8 = 2 * 2 * 2 * 4 = 2 * 2 * 2 * 2 * 2 = 2^5.
- The mixed radical becomes: 3√32 = 3√(2^5) = 3 * 2√(2^2) = 6√(2^2) = 6 * 2 = 12.

5. 2√18:
- Factoring 18: 18 = 2 * 9 = 2 * 3 * 3.
- So, 2√18 = 2√(2 * 3 * 3) = 2 * 3√2 = 6√2.

6. 5√48:
- Factoring 48: 48 = 2 * 24 = 2 * 2 * 12 = 2 * 2 * 2 * 6.
- Simplifying the mixed radical: 5√48 = 5√(2 * 2 * 2 * 6) = 5 * 2√6 = 10√6.

7. 3√54:
- Factoring 54: 54 = 2 * 27 = 2 * 3 * 9 = 2 * 3 * 3 * 3.
- Finally, we can write the mixed radical as: 3√54 = 3√(2 * 3 * 3 * 3) = 3 * 3√2 = 9√2.

I hope this step-by-step explanation helps you understand how to simplify mixed radicals!

To simplify mixed radicals, we need to break down the numbers inside the square root sign into their prime factors. Then, we can move pairs of the same prime factors outside of the square root sign.

Let's simplify each of the mixed radicals you listed:

1. 2√48:
- First, find the prime factorization of 48: 2 * 2 * 2 * 2 * 3 = 2^4 * 3.
- We can take out two pairs of the same prime factors from the square root: 2 * 2 = 4.
- The simplified radical is 4√3.

2. 3√81:
- 81 is already a perfect square (9 * 9 = 81).
- The simplified radical is 9√3.

3. 6√12:
- Find the prime factorization of 12: 2 * 2 * 3 = 2^2 * 3.
- Take out one pair of the same prime factors from the square root: 2.
- The simplified radical is 2√3.

4. 3√32:
- Find the prime factorization of 32: 2 * 2 * 2 * 2 * 2 = 2^5.
- Take out two pairs of the same prime factors from the square root: 2 * 2 = 4.
- The simplified radical is 4√2.

5. 2√18:
- Find the prime factorization of 18: 2 * 3 * 3 = 2 * 3^2.
- Take out one pair of the same prime factors from the square root: 3.
- The simplified radical is 3√2.

6. 5√48:
- Find the prime factorization of 48: 2 * 2 * 2 * 2 * 3 = 2^4 * 3.
- Take out two pairs of the same prime factors from the square root: 2 * 2 = 4.
- The simplified radical is 4√3.

7. 3√54:
- Find the prime factorization of 54: 2 * 3 * 3 * 3 = 2 * 3^3.
- Take out one pair of the same prime factors from the square root: 3.
- The simplified radical is 3√6.

I hope this explanation helps you understand how to simplify mixed radicals.

Divide 8 to the square root of 6/2