Why $2(0.3)^2$ doesn't equal $0.6^2$

Question

Why $2(0.3)^2$ doesn't equal $0.6^2$?

I mean if $0.6 = 2(0.3)$, then why $2(0.3)^2$ doesn't equal $0.6^2$?

I think it is because of the power but I'm not sure about that.

All that I know is that it is confusing. What's the right way to do it or there is no right way?

Answer 1

The reason is similar to the reason $2+3\times4$ is $2+12$ and not $5\times4$. To figure out $2+3\times4$, you have to do the multiplication first, it’s just a rule about how arithmetic works. Another rule, which maybe you didn’t know, is that when you have a power, you have to do that before multiplication, division, addition, or subtraction.

If things are in parentheses, of course, you work out what’s inside the parentheses first, like when $(2+3)\times5=5\times5$, but in your example, $2\times0.3$ isn’t in its own set of parentheses, so you don’t multiply first, you do the power first.

Answer 2

You're doing $$ 2a^2=2\cdot (a\cdot a) $$ and you can't get, in general, $(2a)^2=2a\cdot 2a=4a^2$.

Yours is a notation problem, I guess: the exponent only applies to the term it appears next to, or to a parenthesized formula, like in $(2a)^2$.

By convention, exponents have higher precedence over multiplication.

Answer 3

Since $(ab)^2=a^2b^2\ne ab^2$, you have that

$0.6^2=(2\cdot 0.3)^2=2^2\cdot 0.3^2=4\cdot 0.3^2=0.36\ne 0.18=2\cdot 0.3^2$.