Question

Add or subtract. Simplify by collecting like radical terms if possible.

3√162- √200+ 7√50

Answer was 52√2

Need an idea how they got there

Answer 1

First, in order to simplify the square roots, you need to find the prime factors of 162, 200, and 50.
Do you know how to find the prime factors of a number?

Answer 2

9*18 for 162, 50*4 for 200, 25*2 for 50?

Answer 3

Those products are correct, but they are not the full prime factorizations.
Here they are:
162 = 2 * 3 * 3 * 3 * 3
200 = 2 * 2 * 2 * 5 * 5
50 = 2 * 5 * 5

Answer 4

I see, what would i do next?

Answer 5

Alright I get to that point so far

Answer 6

Since we are dealing with square roots, we need factors that come in multiples of 2.

\(3\sqrt{162} - \sqrt{200} + 7\sqrt{50}\)

\(3 \sqrt{2 \times 3^4} - \sqrt{2^2 \times 2 \times 5^2 } + 7\sqrt{2 \times 5^2}\)

\(3 \sqrt{2} \times \sqrt{3^4} - \sqrt{2^2} \times \sqrt{2} \times \sqrt{5^2 } + 7\sqrt{2} \times \sqrt{5^2}\)

\(3 \sqrt{2} \times \color{red}{\sqrt{81}} - \color{red}{\sqrt{4}} \times \sqrt{2} \times \color{red}{\sqrt{25 }} + 7\sqrt{2} \times \color{red}{\sqrt{25}}\)

Now you see which square roots can be simplified.

Answer 7

Now we take the square roots of 25, 81, and 4.

Answer 8

\(9 \times 3\sqrt{2} - 2 \times 5 \times \sqrt{2} + 5 \times 7\sqrt{2} \)

Answer 9

\(27\sqrt{2} - 10 \times \sqrt{2} + 35\sqrt{2} \)

Answer 10

Now we the terms:

\(52 \sqrt{2} \)

Answer 11

Ah I see thanks