NEW!!! Tough and tricky exponents and roots questions

HOW the second number is much larger then the first one?

Why is second number much larger then the first one? Consider this, even if we had (100^3)^4 (instead of (17^3)^4) and 1000^(3^2) (instead of 1973^(3^2)) --> (100^3)^4=100^12=10^24 and 1000^(3^2)=1,000^9=10^27.

I explained this several times in this thread: https://gmatclub.com/forum/new-tough-an ... l#p2043010

Hope it's clear.

My approach was 26^n is possible with only multiples of 2 and 13. 13 is a prime number and not in the given value of x. thus,n has to zero. Is this correct ?

Yes, x is NOT a multiple of 13, bur 26^n IS if n > 0, so n must be 0.

5+4*5+4*5^2+4*5^3+4*5^4+4*5^5
= 5 + (5 - 1)*5 + (5 - 1)*5^2 + (5 - 1)*5^3 + (5 - 1)*5^4 + (5 - 1)*5^5
= 5 + (5^2 - 5) + (5^3 - 5^2) + (5^4 - 5^3) + (5^5 - 5^4) + (5^6 - 5^5)

Observe that alternate terms cancel out, giving us:

Sum = 5^6

Answer A

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.

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.

Shouldn't the answer be +2sqrt15 or -2sqrt15? An equation with a squared variable always has two solutions. Bunuel TargetTestPrep

So, we get that \((\sqrt{25+10\sqrt{6}}+\sqrt{25-10\sqrt{6}})^2=\sqrt{60}\).

\(\sqrt{25+10\sqrt{6}}+\sqrt{25-10\sqrt{6}}\) is the sum of two square roots, which cannot be negative.

Mathematically, \(\sqrt{...}\) is the square root sign, a function (called the principal square root function), which cannot give negative result. So, this sign (\(\sqrt{...}\)) always means non-negative square root.


The graph of the function f(x) = √x

Notice that it's defined for non-negative numbers and is producing non-negative results.

TO SUMMARIZE:
When the GMAT (and generally in math) provides the square root sign for an even root, such as a square root, fourth root, etc. then the only accepted answer is the non-negative root. That is:

\(\sqrt{9} = 3\), NOT +3 or -3;
\(\sqrt[4]{16} = 2\), NOT +2 or -2;

Notice that in contrast, the equation \(x^2 = 9\) has TWO solutions, +3 and -3. Because \(x^2 = 9\) means that \(x =-\sqrt{9}=-3\) or \(x=\sqrt{9}=3\).

Hope it helps.

please add more questions like this... not only in this session but please add this type of questions in other topics like fractions, profits etc...

Check the list below:

PS:

• Tough and tricky exponents and roots questions
• 12 Easy Pieces (or not?)
• Bakers' Dozen
• Algebra set
• Mixed Questions
• Fresh Meat
• New Year's Quantitative Challenge Set: Jedi Edition!!! - Challenge yourself with these Hand Picked Tricky Problem Solving and Data Sufficiency Questions.
• Bunuel's Algebra PS Diagnostic Test
• Bunuel's Word Problems PS Diagnostic Test

DS:

• Tough and tricky exponents and roots questions
• The Discreet Charm of the DS
• Devil's Dozen!!!
• Number Properties set
• New DS set
• Bunuel's Algebra DS Diagnostic Test
• Bunuel's Word Problems DS Diagnostic Test

Hope it helps.

Q7. This is actually very easy. Notice that, 3^2 is 9, 3^3 is 2.
First term > 3^2 i.e. 9, so we can say that the root is obviously greater than 3. Same goes with the next term, 3√9 > 2^3, so it is greater than 2 aswell. For the rest of the terms, they are all greater than 1. If we add every element, it will always be greater than 12.
Hence, option E

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In the last line I did Units digit for xxx1 - xxxx3 = 8, Can you help what am i doing wrong.
[Reference -You mentioned unit digit will be 2 as 1-2= -2]

I think your doubt is addressed in the solution you quite. Here it is again in more details:

What is the units digit of \((17^3)^4-1973^{3^2}\)?

A. 0
B. 2
C. 4
D. 6
E. 8


MUST KNOW FOR THE GMAT:

I. The units digit of \((abc)^n\) is the same as that of \(c^n\).

II. \({(a^m)}^n=a^{mn}\) and \(a^{m^n}=a^{(m^n)}\).

III. The units digit of integers in positive integer powers repeats in a specific pattern called cyclicity. The units digit of 7 and 3 in positive integer powers repeats in patterns of 4:

Hence, the units digit of \(17^{12}\) is 1, as the 12th number in the repeating pattern 7-9-3-1 is 1, and the units digit of \(1973^9\) is 3, as the 9th number in the repeating pattern 3-9-7-1 is 3.

BACK TO THE QUESTION:

Thus, we know that the units digit of \({(17^3)}^4=17^{12}\) is 1, and the units digit of \(1973^{3^2}=1973^9\) is 3. If the first number is greater than the second number, the units digit of their difference will be 8, for example, 11 - 3 = 8. However, if the second number is greater than the first number, the units digit of their difference will be 2, for example, 11 - 13 = -2.

Consider this, even if the first number were \(100^{12}\) instead of \(17^{12}\), and the second number were \(1000^9\) instead of \(1973^9\), the first number, \(100^{12}=10^{24}\), would still be smaller than the second number \(1,000^9=10^{27}\). Therefore, \(17^{12} < 1973^9\), and the units digit of the difference is 2.


Answer: B

Hi, can anyone explain why we need to un-square the value? What is the rule or logic behind this?

Thank you.

Hello dendenden

Below is detailed answer. I hope it helps!

Squaring an expression containing square roots is a common algebraic technique used to eliminate the square roots and simplify the expression. Conversely, taking the square root is an operation performed to balance the initial squaring. Essentially, first we square to simplify and eliminate the square roots but at the end to we get the squared value of the expression, and to get the original value, we need to unsquare.

Below is more detailed solution, which might help:


What is the value of \( \sqrt{25 + 10 \sqrt{6} } + \sqrt{ 25 - 10 \sqrt{6} }\) ?


A. \(2\sqrt{5}\)
B. \(\sqrt{55}\)
C. \(2\sqrt{15}\)
D. 50
E. 60


First, square the given expression to eliminate the square roots, but remember to take the square root of the result at the end to balance the operation and obtain the correct answer.

Important for the GMAT: \((x+y)^2=x^2+2xy+y^2\) and \((x-y)^2=x^2-2xy+y^2\).

Following these rules, we get:

Note that the sum of the first and third terms simplifies to \((25+10\sqrt{6})+(25-10\sqrt{6})=50\), so we have:

Another important concept for the GMAT: \((x+y)(x-y)=x^2-y^2\). Using this, we can simplify further:

Finally, remember to take the square root of this value to obtain the correct answer: \(\sqrt{60}=2\sqrt{15}\).


Answer: C