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Exponents and roots problems are very common on the GMAT. So, it's extremely important to know how to manipulate them, how to factor out, take roots, multiply, divide, etc. Below are 11 problems to test your skills. Please post your thought process/solutions along with the answers.

I'll post OA's with detailed solutions tomorrow. Good luck.

1. 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

5. If \(x=23^2*25^4*27^6*29^8\) and is a multiple of \(26^n\), where \(n\) is a non-negative integer, then what is the value of \(n^{26}-26^n\)? A. -26 B. -25 C. -1 D. 0 E. 1

7. If \(x=\sqrt{10}+\sqrt[3]{9}+\sqrt[4]{8}+\sqrt[5]{7}+\sqrt[6]{6}+\sqrt[7]{5}+\sqrt[8]{4}+\sqrt[9]{3}+\sqrt[10]{2}\), then which of the following must be true: A. x<6 B. 6<x<8 C. 8<x<10 D. 10<x<12 E. x>12

8. If \(x\) is a positive number and equals to \(\sqrt{6+{\sqrt{6+\sqrt{6+\sqrt{6+...}}}}}\), where the given expression extends to an infinite number of roots, then what is the value of x? A. \(\sqrt{6}\) B. 3 C. \(1+\sqrt{6}\) D. \(2\sqrt{3}\) E. 6

9. If \(x\) is a positive integer then the value of \(\frac{22^{22x}-22^{2x}}{11^{11x}-11^x}\) is closest to which of the following? A. \(2^{11x}\) B. \(11^{11x}\) C. \(22^{11x}\) D. \(2^{22x}*11^{11x}\) E. \(2^{22x}*11^{22x}\)

10. Given that \(5x=125-3y+z\) and \(\sqrt{5x}-5-\sqrt{z-3y}=0\), then what is the value of \(\sqrt{\frac{45(z-3y)}{x}}\)? A. 5 B. 10 C. 15 D. 20 E. Can not be determined

Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]

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12 Jan 2012, 12:36

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5. If x=23^2*25^4*27^6*29^8 and is a multiple of 26^n, where n is a non-negative integer, then what is the value of n^26-26^n? A. -26 B. -25 C. -1 D. 0 E. 1

Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]

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12 Jan 2012, 12:51

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8. If x is a positive number and equals to \sqrt{6+{\sqrt{6+\sqrt{6+\sqrt{6+...}}}}}, where the given expression extends to an infinite number of roots, then what is the value of x? A. \sqrt{6} B. 3 C. 1+\sqrt{6} D. 2\sqrt{3} E. 6

x= √(6+√(6+√(6+... x^2 = 6 + x x^2-x-6=0 (x-3)(x+2) = 0 x = 3 or -2 Value of sqrt cant be -ve Option

Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]

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12 Jan 2012, 13:01

10. Given that 5x=125-3y+z and \sqrt{5x}-25-\sqrt{z-3y}=0, then what is the value of \sqrt{\frac{45(z-3y)}{x}}? A. 5 B. 10 C. 15 D. 20 E. Can not be determined

5x-125=z-3y

sqrt{5x}-25=\sqrt{z-3y} (sqrt{5x}-25)^2=z-3y

(sqrt{5x}-25)^2=5x-125 \sqrt{5x}=15 x=5*9=45

so the expression \sqrt{\frac{45(z-3y)}{x} = \sqrt{45(15-25)/45}=\sqrt{-10}

E is the answ
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Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]

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12 Jan 2012, 13:07

7. If x=\sqrt{10}+\sqrt[3]{9}+\sqrt[4]{8}+\sqrt[5]{7}+\sqrt[6]{6}+\sqrt[7]{5}+\sqrt[8]{4}+\sqrt[9]{3}+\sqrt[10]{2}, then which of the following must be true: A. x<6 B. 6<x<8 C. 8<x<10 D. 10<x<12 E. x>12

my answ to this q is intuitive.

everyone knows that sqroot10=3.16 \sqrt[3]{9} is approx \sqrt[3]{2^3} >2 (but less than 3) the rest of roots are more than 1 and less than 2

so we have 3.16+2,..+1+1+1+1+1+1+1 = approx 12.16

E is the answ
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I am still on all gmat forums. msg me if you want to ask me smth

Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]

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12 Jan 2012, 13:17

9. If x is a positive integer then the value of \frac{22^{22x}-22^{2x}}{11^{11x}-11^x} is closest to which of the following? A. 2^{11x} B. 11^{11x} C. 22^{11x} D. 2^{22x}*11^{11x} E. 2^{22x}*11^{22x}

3. If \(5^{10x}=4,900\) and \(2^{\sqrt{y}}=25\) what is the value of \(\frac{5^{(x-1)^5}}{4^{-\sqrt{y}}\)? A. 14/5 B. 5 C. 28/5 D. 13 E. 14

5. If \(x=23^2*25^4*27^6*29^8\) and is a multiple of \(26^n\), where \(n\) is a non-negative integer, then what is the value of \(n^26-26^n\)? A. -26 B. -25 C. -1 D. 0 E. 1

Welcome back Bunuel!

Some nice questions. A couple of comments:

Q5 has a typo at the end: \(n^26-26^n\). I think you need to include the exponent 26 in braces {26} to get \(n^{26}-26^n\)

In Q3, unless I'm making some kind of stupid mistake, the value of the expression in the question is close to 600 (so isn't among the answers). I'm wondering if you meant the question to instead read something more like:

3. If \(5^{10x}=4,900\) and \(4^{\sqrt{y}}=25\) what is the value of \(\frac{5^{5x-1}}{2^{\sqrt{y}}\)? A. 14/5 B. 5 C. 28/5 D. 13 E. 14

In that case, I get one of the five answer choices (A).
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9. If x is a positive integer then the value of \frac{22^{22x}-22^{2x}}{11^{11x}-11^x} is closest to which of the following? A. 2^{11x} B. 11^{11x} C. 22^{11x} D. 2^{22x}*11^{11x} E. 2^{22x}*11^{22x}

And it looks as though every other question has been solved except for this one. I won't solve it completely, but I'll help people get started. Here, since we are asked for an approximation only, we don't need to compute an exact value. You might notice that if x is a positive integer, \(22^{22x}\) is vastly bigger than \(22^{2x}\). Since that's true, if we only need to estimate, we can ignore the small term. We can do the same thing in the denominator.
_________________

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If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com

Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]

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12 Jan 2012, 13:48

IanStewart wrote:

And it looks as though every other question has been solved except for this one. I won't solve it completely, but I'll help people get started. Here, since we are asked for an approximation only, we don't need to compute an exact value. You might notice that if x is a positive integer, \(22^{22x}\) is vastly bigger than \(22^{2x}\). Since that's true, if we only need to estimate, we can ignore the small term. We can do the same thing in the denominator.

And it looks as though every other question has been solved except for this one. I won't solve it completely, but I'll help people get started. Here, since we are asked for an approximation only, we don't need to compute an exact value. You might notice that if x is a positive integer, \(22^{22x}\) is vastly bigger than \(22^{2x}\). Since that's true, if we only need to estimate, we can ignore the small term. We can do the same thing in the denominator.

If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com

3. If \(5^{10x}=4,900\) and \(2^{\sqrt{y}}=25\) what is the value of \(\frac{5^{(x-1)^5}}{4^{-\sqrt{y}}\)? A. 14/5 B. 5 C. 28/5 D. 13 E. 14

5. If \(x=23^2*25^4*27^6*29^8\) and is a multiple of \(26^n\), where \(n\) is a non-negative integer, then what is the value of \(n^26-26^n\)? A. -26 B. -25 C. -1 D. 0 E. 1

Welcome back Bunuel!

Some nice questions. A couple of comments:

Q5 has a typo at the end: \(n^26-26^n\). I think you need to include the exponent 26 in braces {26} to get \(n^{26}-26^n\)

In Q3, unless I'm making some kind of stupid mistake, the value of the expression in the question is close to 600 (so isn't among the answers). I'm wondering if you meant the question to instead read something more like:

3. If \(5^{10x}=4,900\) and \(4^{\sqrt{y}}=25\) what is the value of \(\frac{5^{5x-1}}{2^{\sqrt{y}}\)? A. 14/5 B. 5 C. 28/5 D. 13 E. 14

In that case, I get one of the five answer choices (A).

Hello Ian. Glad to be back!

Thanks for your comments. Yes, there were 2 typos: For Q5: included the exponent 26 in braces {26} to avoid confusion; For Q3: added the brackets which were lost in copy/past, it should read \(\frac{(5^{(x-1)})^5}{4^{-\sqrt{y}}\).
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Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]

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13 Jan 2012, 14:39

LalaB wrote:

2. What is the units digit of (17^3)^4-1973^{3^2}? A. 0 B. 2 C. 4 D. 6 E. 8

7^12-3^9 11-3=8

ANS IS E

Bunuel wrote:

2. What is the units digit of \((17^3)^4-1973^{3^2}\)? A. 0 B. 2 C. 4 D. 6 E. 8

i believe the answer is : 7^12-3^6 ---> 11-9= 2 ---> B

note: I believe the part I wrote (which i colored red) is wrong. Can someone explain why? when an exponent is raised to another exponent then what operation should we do first? - Okey! Just read the rules about routs that Bunuel posted and found my mistake! (exponentiation by stacked symbols)

Last edited by SonyGmat on 14 Jan 2012, 00:11, edited 6 times in total.

Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]

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13 Jan 2012, 16:09

LalaB wrote:

10. Given that 5x=125-3y+z and \sqrt{5x}-25-\sqrt{z-3y}=0, then what is the value of \sqrt{\frac{45(z-3y)}{x}}? A. 5 B. 10 C. 15 D. 20 E. Can not be determined

5x-125=z-3y

sqrt{5x}-25=\sqrt{z-3y} (sqrt{5x}-25)^2=z-3y

(sqrt{5x}-25)^2=5x-125 \sqrt{5x}=15 x=5*9=45

so the expression \sqrt{\frac{45(z-3y)}{x} = \sqrt{45(15-25)/45}=\sqrt{-10}

E is the answ

bunuel wrote:

10. Given that \(5x=125-3y+z\) and \(\sqrt{5x}-25-\sqrt{z-3y}=0\), then what is the value of \(\sqrt{\frac{45(z-3y)}{x}}\)? A. 5 B. 10 C. 15 D. 20 E. Can not be determined

Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]

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13 Jan 2012, 19:42

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IanStewart wrote:

Bunuel wrote:

3. If \(5^{10x}=4,900\) and \(2^{\sqrt{y}}=25\) what is the value of \(\frac{5^{(x-1)^5}}{4^{-\sqrt{y}}\)? A. 14/5 B. 5 C. 28/5 D. 13 E. 14

5. If \(x=23^2*25^4*27^6*29^8\) and is a multiple of \(26^n\), where \(n\) is a non-negative integer, then what is the value of \(n^26-26^n\)? A. -26 B. -25 C. -1 D. 0 E. 1

Welcome back Bunuel!

Some nice questions. A couple of comments:

Q5 has a typo at the end: \(n^26-26^n\). I think you need to include the exponent 26 in braces {26} to get \(n^{26}-26^n\)

In Q3, unless I'm making some kind of stupid mistake, the value of the expression in the question is close to 600 (so isn't among the answers). I'm wondering if you meant the question to instead read something more like:

3. If \(5^{10x}=4,900\) and \(4^{\sqrt{y}}=25\) what is the value of \(\frac{5^{5x-1}}{2^{\sqrt{y}}\)? A. 14/5 B. 5 C. 28/5 D. 13 E. 14

In that case, I get one of the five answer choices (A).

then \(\sqrt{25+10\sqrt{6}}*\sqrt{25-10\sqrt{6}}=y\) So \(({25+10\sqrt{6}})*({25-10\sqrt{6}})=y^2\) so \(625-600=y^2\) so \(y=5\) so \(2ab=2y=10\) so \(x^2=a^2+b^2+2ab=50+10=60\) so \(x^2=60\) so \(x=\sqrt{4*15}\) so \(x=2\sqrt{15}\)

Hence Answer is C. \(2\sqrt{15}\)
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