Subtracting expressions with radicals

Question

I want to subtract the expressions $20\sqrt{72a^3b^4c} - 14\sqrt{8a^3b^4c}$. I simplified this to $120ab^2\sqrt{2ac}-28ab^2\sqrt{2ac}$. My textbook says the answer is $92ab^2\sqrt{2ac}$. Why doesnt the $ab^2\sqrt{2ac}$ part change at all? I thought everything except the 92 would cancel out since it looks like it's cancelling out. This is my first time using stackexchange, please tell me if I can ask this question better. Thanks.

Answer 1

Just view $ab^2\sqrt{2ac}$ as a separate entity.

As a matter of fact, let us use $d$ to represent it.

Then, we would have $120d-28d$, which is clearly $92d$.

Answer 2

Observe that $\;120-28=92\;$ , and thus you have an expression of the form

$$120 K-28 K=92K\,.\;\text{In this case, we simply have}\;\;K=ab^2\sqrt{2ac}\;$$

Answer 3

Welcome to the site.

you have: $120ab^2\sqrt{2ac}-28ab^2\sqrt{2ac}$

What is the highest power of $a,b,c$ and rational numbers common to both terms? Factor that out

$a: a\sqrt a\\ b: b^2\\ c: \sqrt c\\ \mathbb N: 4\\ 4ab^2\sqrt{ac}(30 - 7)\\ 4ab^2\sqrt{ac}(23)\\ 92ab^2\sqrt{ac}$

Answer 4

The $ab^2\sqrt{2ac} $ is a common factor between the two terms.

This means you can rewrite the whole expression as: $ ab^2\sqrt{2ac} (120 - 28) $.

Then you simplify what is inside the parenthesis and you get the answer your book has.

Answer 5

Although this question has several good answers and good points in the comments, I want to point out the explicit connection to the distributive law, in terms that can help in elementary school classrooms too.

Kids have little trouble answering the question

If I have 7 apples and get 2 more how many do I have then?

If you want to write the answer with more formal mathematics (nowadays called a number sentence) it's

7 apples + 2 apples = 9 apples .

This principle helps kids master place value: think "hundreds" instead of "apples" and you get $700 + 200 = 900$, even $900 + 200 = 1100$.

You can even touch on the etymology of "ninety" as coming from "nine tens".

In the OP's question the unit quantity of which there are first 128 and then only 92 is $ab^2\sqrt{2ac}$.