andreagonzalez2k wrote:
A question about chained exponents. In the second question you say:
\({1973^{3}}^{2} = 1973^{9}\)
But, as the '3' digit is not bigger in font size than the '2' digit I interpreted it as:
\({1973^{3}}^{2} = 1973^{6}\)
and my interpretacion is consistent with the way we write mathematical formulas here:
{1973^{3}}^{2} (look at the braces)
For example, if you have:
\({{{{7^{2}}^{3}}^{4}}^{5}}\)
How should we interpret it?
The font size actually doesn't matter. The only thing that matters is whether there are parentheses. When there are no parentheses, as in your examples, we always apply the exponents from the top down. So
\(
7^{2^5} = 7^{32}
\)
(with no brackets, we always interpret it to mean \(7^{(2^5)}\)) and is not equal to
\(
(7^2)^5 = 7^{10}
\)
You actually don't see the first situation about very often on the GMAT -- it's rare (but not impossible) that you'll see something like
\(
2^{3^2}
\)
(which equals 2^9) in a GMAT question. Usually when I've seen something this situation on the GMAT, there's an unknown in the exponent, e.g.:
\(
5^{x^2}
\)
(which is not the same thing as \(5^{2x}\).) The situation with parentheses, though, is one that comes up all the time when you're solving exponent questions, so it's likely what you've encountered most often.
And the number you ask about, \({{{{7^{2}}^{3}}^{4}}^{5}}\), might look harmless enough, but it is absurdly enormous. Just the 4^5 part is already roughly 1000, then we raise 3 to that power to get something close to 10^500. Then we need to raise the 2 to that power, and raise the 7 to whatever absurdly large number we get. That's not something you'd ever need to deal with on the GMAT, fortunately.
I know the example I have chosen is not idoneus. But with smaller numbers it could appear.
Without parenthesis you have to know what is the default interpretation. It would be clearer to put parenthesis always: