And don’t discount No Child Left Behind. If nothing else, it put qualified teachers into middle school math and science classes taking over for teachers that did not know math and science.
They should at least have taught that a root is the reciprocal of an exponent.
I would just say roots are fractional exponents. But it’s never something I thought about in terms of order of operations because I always thought of roots as being exponents.
I don’t understand how there could be any ambiguity about roots, if we’re talking about using a radical sign. The vinculum at the top of the sign shows what part of the expression is included. It’s like built-in parentheses. For example, in (\sqrt{4x}\:y), the 4x is raised to the power \frac{1}{2} but the y isn’t, because it’s not under the vinculum. What would be an example of an expression in which a root operation could be interpreted in more than one way?
Right. But too often, students get careless about what is and is not under the radical sign.
Sure, students get careless about a lot of things. But it seems to me the solution is to train them to use a radical sign properly, rather than to invent a fake rule about “root precedence” that may lead them into making errors in either reading or constructing equations.
But what if there’s no vinculum, and there doesn’t need to be. How would you value \sqrt{} 4 9? is it 2x9 = 18 or 7? I’ll admit that it’s poorly written, but I think technically 18 is correct.
And does it make a difference how big a space I leave between the 4 and the 9?
√49 is only 18 if 49 is 36.
The multiply sign can be skipped between single-letter variables, expressions, or a string of digit(s) next to a variable or expression.
It cannot be skipped between digits. “12” is twelve. “1 2” is undefined.
If you read it as square root of 4 times 9 it’s 18. I’m not saying I read it that way, but was just trying to talk about the need for (or not) a vinculum.
My view is that a radical sign without a vinculum is malformed. There’s no more need to try to understand how to read it than there is to read an expression with unbalanced parentheses. But that’s just how I was taught; maybe in some circles it’s considered ok to omit the vinculum.
I agree. I don’t know why you’d write one that way unless it was in a context where you couldn’t write it properly, like typing individual characters on a line. And in that case, you should use parentheses where necessary to make it clear what the radicand consists of.
And 49 is 36 if you read it as 4 times 9.
Even if you must use a radical sign without a vinculum, it only makes sense to interpret it as applying to the next number, not merely the next digit. I suppose that that’s a binding even earlier in the order of operations than the ones everyone discusses: Formation of place-value numbers via concatenation of digits takes precedence over everything.
How about this one?
-92 - (-9)2 = ?
In my mind:
-(9 x 9) - (-9 x -9) =
-81 - 81 = -162
What questions do you have? You may want to note that for any number a you have (-a)^2=a^2, therefore -a^2-(-a^2)=-2a^2.
Yup, that one is pretty standard. The exponent doesn’t apply to the negative sign (nor to any other multiplication) unless the negative sign is in parentheses.
A lot of people are coming up with zero. It’s obvious to me that the first amount is the negative value of 92, but there are people saying 'when you square a negative, it’s always positive. Which is true; only we’re not squaring a negative. We’re making a square negative.
But since what was obvious to me in the OP turned out to be ambiguous; so I was wondering if there was ambiguity in this equation.
It shouldn’t be ambiguous but, notoriously, Excel gets this wrong. If you type “=-9^2” into an Excel cell, it will give you 81, not -81. You must use parentheses to convince Excel to do it correctly.
Don’t get me started on Excel!
Ouch, you’re right. (And Google Sheets does it too. Morons!)