## To simplify the expression v45 + v80 - 3v20, you want to simplify each individual radical separately and then combine like terms.

Let's start with each term:

- v45 represents the square root of 45. To simplify this, you can break it down into a product of prime factors: 45 = 9 x 5. Since the square root of 9 is 3 and the square root of 5 cannot be simplified further, you get v45 = 3v5.

- v80 represents the square root of 80. Similarly, you can factorize 80 as 16 x 5. The square root of 16 is 4, so v80 = 4v5.

- 3v20 represents the square root of 20 multiplied by 3. Factoring 20 gives you 4 x 5. The square root of 4 is 2, so 3v20 = 6v5.

Now, you can substitute these simplified terms back into the original expression:

3v5 + 4v5 - 6v5

Since all three terms have the same radical term (v5), you can simply combine the coefficients:

3v5 + 4v5 - 6v5 = (3 + 4 - 6)v5 = v5

Therefore, the simplified expression is just v5.

Moving on to the next expression, (3v5 + v3)(v2 + 2v5), you can use the distributive property to expand it:

(3v5 + v3)(v2 + 2v5) = 3v5 * v2 + 3v5 * 2v5 + v3 * v2 + v3 * 2v5

Simplifying each term:

- 3v5 * v2 = 3v10 (since you can multiply the coefficients and add the exponents of the radicals when multiplying)

- 3v5 * 2v5 = 6v25 (following the same rule as above, multiplying the coefficients and adding the exponents)

- v3 * v2 = v6 (same rules apply here)

- v3 * 2v5 = 2v15 (again, multiplying the coefficients and adding the exponents)

Now we can combine like terms:

3v10 + 6v25 + v6 + 2v15

Notice that v10, v25, v6, and v15 all have unique radical terms, so there is no further simplification possible. This is the simplified expanded form of the expression.

Finally, for the expression (v3 + 4v7)², you can square each term within the parentheses:

(v3 + 4v7)² = (v3)² + 2(v3)(4v7) + (4v7)²

Simplifying each term:

- (v3)² = v6 (since the square cancels out the radical sign)

- 2(v3)(4v7) = 8v21 (you can multiply the coefficients and add the exponents)

- (4v7)² = 16v49 (following the same rule as above)

Now combine the like terms:

v6 + 8v21 + 16v49

Again, notice that each term has a unique radical term, so this is the simplified expanded form of the expression.

As for the equation �ãx-1=3, it seems to be a separate question or equation unrelated to the previous expressions. To solve it, you want to isolate the variable x:

�ãx-1=3

First, you can add 1 to both sides of the equation to eliminate the -1 term:

�ãx = 3+1

�ãx = 4

Now, to get rid of the square root, you can square both sides of the equation:

(�ãx)² = 4²

x = 16

So the solution to the equation �ãx-1=3 is x = 16.

I hope this explanation helps!