Combine the radicals, I wasn't sure what to do with the √7, or how to simplify it, i need a brief explanation

Question

Combine the radicals

4√7+3√28

Answer 1

The key is to place in simplest radical form the √28. Think of rewriting the 28 as 4 times 7. Then take the square root of 4. So √28 becomes 2√7.

Now let's go back to the original expression replacing the √28 with its simplest form:

4 √7 + 3 (2 √7) = 4 √7 + 6 √7. These two terms can be added to get 10 √7. To see why:

What if you had 4A + 6A. Picture factoring out the A to get A (4 + 6) or 10A. Same idea using the common factor √7.

Answer 2

square root of 7 can not be simplified further (7 is a prime number)

however, square root of 28 = square root (7*4) = square root(4) * square root(7) = 2*square root of 7.

So to combine the radicals:

4√7+3√28

= 4√7+3* 2*√7

= 4√7+6√7

= 10√7

Answer 3

Applying two key rules here helps us simplify this expression.

First, the (positive) square root of the product of two numbers is the product of the (positive) square roots of the same numbers. In algebraic terms:

√(xy) = (√x)(√y)

Second, the (positive) square root of the square of a non-negative number is that number. This is evident from the definition of square root: the number that, when squared, gives us back the original number. In algebraic terms:

√(a²) = a for all a ≥ 0

Unfortunately, √7 cannot be simplified using the rules above (or by any other technique). This is because 7 is a prime number, meaning that its only factors are itself and one. It's fine to leave an expression in its simplest possible form.

We can factor 28 as 2²·7 and then apply the rules above to simplify the 3√28 term. Done correctly, it should simplify to an integer multiplied by √7. Now, you should have an expression which looks like this, where the blank space is an integer:

4√7 + _√7

This expression can be simplified by applying the distributive property, which allows us to combine like terms using this formula:

ac + bc = (a+b)c

N.B. The two answers previously provided also work, but this answer provides a conceptual understanding of why they work.