Hello,
Are these 700 questions ? i'm kinda struggling with them.

Yes, those are very difficult questions. Probably even 800.
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I am absolutely doomed if GMAT is even half as difficult

Posted from my mobile device
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Some of the questions here are more difficult than anything you'd ever see on the real test. The questions above are mostly for practice. Check easier questions using our question bank's filters: https://gmatclub.com/forum/search.php?view=search_tags
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I think we should consider this way..
for 7^12 units place is 7 as below...

7*7=49
9*7=63
3*7=21
1*7=7
..........
7*7=49
9*7=63... so repeating (so for 12 it will be 7.)

next for 3^9 units place is 9 as below...

3*3=9
9*3=27
7*3=21
1*3=3
.........
3*3=9
9*3=27... so repeating (so for 9 it will be 9)

Hence answer is 9-7 is 2.

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?

There are a lot of different interpretations and results.

I think your doubt is addressed in the solution of this question HERE as well as HERE.
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So

\({{{{7^{2}}^{3}}^{4}}^{5}}\)

should be interpreted always as:

- \(4^{5} = 1024\)

- \(3^{1024} = a\)

- \(2^{a} = b\)

- and finally \(7^{b}\)

?

________________________________
Yes.
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Thanks a lot for your answer. I think the formula would be clearer if all the exponents in the example were in descending font size as in handwritten formulas, but it is OK.

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.
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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:

\(7^{(a^{(b^{(c^{d})})})}\)

Answer for the 7th Question:
1st term : square root of 10>3
2nd term : cube root of 9> cube root of 8=2
rest all terms are greater than 1
therefore required sum; x>3+2+1+1+1+1+1+1+1

Bunuel chetan2u

I took a bit different approach to this problem and got an incorrect answer. Can you please help me understand the flaw in my approach. I am unable to understand where have I gone wrong.

You are wrong while substituting 3 in 1.
5x=125-3y+z=125+z-3y=125+100=225,
x=225/5=45
Now you will get your answer.
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Thanks for the prompt response ! Didn't realize this silly error

Great questions Bunuel!
Is the following thinking correct or should I adjust some of my logic?
1) I see som square roots, so I should probably square both sides...
2) The whole thing has to be positive, since the square root of a number is never negative.
3) I can then square it with no problem because I know the value is positive.
4) If the sign between the square roots was - and not +, I could not automatically square it because I would not know if it was positive.
5) If It had a negative constant as well (like \(\sqrt{25+10\sqrt{6}}+\sqrt{25-10\sqrt{6}}-6\), I could not automatically square it because I do not know if the total value is positive. But if this were the case, I could make the whole thing an equation and introduce a variable on the RHS (for the value we are looking for), move the constant over to the RHS, square both sides, find the root/roots and check if they in fact fit in the original made equation or of they are extraneous roots.

I would love to some thoughts on this.