NEW!!! Tough and tricky exponents and roots questions : GMAT Data Sufficiency (DS) - Page 2
Check GMAT Club Decision Tracker for the Latest School Decision Releases http://gmatclub.com/AppTrack

 It is currently 23 Jan 2017, 18:30

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# NEW!!! Tough and tricky exponents and roots questions

Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 36618
Followers: 7100

Kudos [?]: 93579 [18] , given: 10578

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

### Show Tags

12 Jan 2012, 02:50
18
KUDOS
Expert's post
99
This post was
BOOKMARKED
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. If $$357^x*117^y=a$$, where $$x$$ and $$y$$ are positive integers, what is the units digit of $$a$$?
(1) $$100<y^2<x^2<169$$
(2) $$x^2-y^2=23$$

Solution: tough-and-tricky-exponents-and-roots-questions-125967.html#p1029239

2. If x, y, and z are positive integers and $$xyz=2,700$$. Is $$\sqrt{x}$$ an integer?
(1) $$y$$ is an even perfect square and $$z$$ is an odd perfect cube.
(2) $$\sqrt{z}$$ is not an integer.

Solution: tough-and-tricky-exponents-and-roots-questions-125967.html#p1029240

3. If $$x>y>0$$ then what is the value of $$\frac{\sqrt{2x}+\sqrt{2y}}{x-y}$$?
(1) $$x+y=4+2\sqrt{xy}$$
(2) $$x-y=9$$

Solution: tough-and-tricky-exponents-and-roots-questions-125967.html#p1029241

4. If $$xyz\neq{0}$$ is $$(x^{-4})*(\sqrt[3]{y})*(z^{-2})<0$$?
(1) $$\sqrt[5]{y}>\sqrt[4]{x^2}$$
(2) $$y^3>\frac{1}{z{^4}}$$

Solution: tough-and-tricky-exponents-and-roots-questions-125967.html#p1029242

5. If $$x$$ and $$y$$ are negative integers, then what is the value of $$xy$$?
(1) $$x^y=\frac{1}{81}$$
(2) $$y^x=-\frac{1}{64}$$

Solution: tough-and-tricky-exponents-and-roots-questions-125967.html#p1029243

6. If $$x>{0}$$ then what is the value of $$y^x$$?
(1) $$\frac{4^{(x+y)^2}}{4^{(x-y)^2}}=128^{xy}$$
(2) $$x\neq{1}$$ and $$x^y=1$$

Solution: tough-and-tricky-exponents-and-roots-questions-125967.html#p1029244

7. If $$x$$ is a positive integer is $$\sqrt{x}$$ an integer?
(1) $$\sqrt{7*x}$$ is an integer
(2) $$\sqrt{9*x}$$ is not an integer

Solution: tough-and-tricky-exponents-and-roots-questions-125967-20.html#p1029245

8. What is the value of $$x^2+y^3$$?
(1) $$x^6+y^9=0$$
(2) $$27^{x^2}=\frac{3}{3^{3y^2+1}}$$

Solution: tough-and-tricky-exponents-and-roots-questions-125967-20.html#p1029246

9. If $$x$$, $$y$$ and $$z$$ are non-zero numbers, what is the value of $$\frac{x^3+y^3+z^3}{xyz}$$?
(1) $$xyz=-6$$
(2) $$x+y+z=0$$

Solution: tough-and-tricky-exponents-and-roots-questions-125967-20.html#p1029247

10. If $$x$$ and $$y$$ are non-negative integers and $$x+y>0$$ is $$(x+y)^{xy}$$ an even integer?
(1) $$2^{x-y}=\sqrt[(x+y)]{16}$$
(2) $$2^x+3^y=\sqrt[(x+y)]{25}$$

Solution: tough-and-tricky-exponents-and-roots-questions-125967-20.html#p1029248

11. What is the value of $$xy$$?
(1) $$3^x*5^y=75$$
(2) $$3^{(x-1)(y-2)}=1$$

Solution: tough-and-tricky-exponents-and-roots-questions-125967-20.html#p1029249
_________________
Math Expert
Joined: 02 Sep 2009
Posts: 36618
Followers: 7100

Kudos [?]: 93579 [5] , given: 10578

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

### Show Tags

14 Jan 2012, 14:46
5
KUDOS
Expert's post
4
This post was
BOOKMARKED
7. If $$x$$ is a positive integer is $$\sqrt{x}$$ an integer?
(1) $$\sqrt{7*x}$$ is an integer
(2) $$\sqrt{9*x}$$ is not an integer

Must know for the GMAT: if $$x$$ is a positive integer then $$\sqrt{x}$$ is either a positive integer itself or an irrational number. (It can not be some reduced fraction eg 7/3 or 1/2)

Also note that the question basically asks whether $$x$$ is a perfect square.

(1) $$\sqrt{7*x}$$ is an integer --> $$x$$ can not be a perfect square because if it is, for example if $$x=n^2$$ for some positive integer $$n$$ then $$\sqrt{7x}=\sqrt{7n^2}=n\sqrt{7}\neq{integer}$$. Sufficient.

(2) $$\sqrt{9*x}$$ is not an integer --> $$\sqrt{9*x}=3*\sqrt{x}\neq{integer}$$ --> $$\sqrt{x}\neq{integer}$$. Sufficient.

_________________
Math Expert
Joined: 02 Sep 2009
Posts: 36618
Followers: 7100

Kudos [?]: 93579 [5] , given: 10578

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

### Show Tags

14 Jan 2012, 14:49
5
KUDOS
Expert's post
5
This post was
BOOKMARKED
8. What is the value of $$x^2+y^3$$?
(1) $$x^6+y^9=0$$
(2) $$27^{x^2}=\frac{3}{3^{3y^2+1}}$$

(1) $$x^6+y^9=0$$ --> $$(x^2)^3=(-y3)^3$$ ---> $$x^2=-y^3$$ --> $$x^2+y^3=0$$. Sufficient.

(2) $$27^{x^2}=\frac{3}{3^{3y^2+1}}$$ --> $$3^{3x^2}=\frac{3}{3^{3y^2}*3}$$ --> $$3^{3x^2}*3^{3y^2}=1$$ --> $$3^{3x^2+3y^2}=1$$ --> $$3x^2+3y^2=0$$ (the power of 3 must be zero in order this equation to hold true) --> $$x^2+y^2=0$$ the sum of two non-negative values is zero --> both $$x$$ and $$y$$ must be zero --> $$x=y=0$$ --> $$x^2+y^3=0$$. Sufficient.

_________________
Math Expert
Joined: 02 Sep 2009
Posts: 36618
Followers: 7100

Kudos [?]: 93579 [9] , given: 10578

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

### Show Tags

14 Jan 2012, 14:52
9
KUDOS
Expert's post
9
This post was
BOOKMARKED
9. If $$x$$, $$y$$ and $$z$$ are non-zero numbers, what is the value of $$\frac{x^3+y^3+z^3}{xyz}$$?
(1) $$xyz=-6$$
(2) $$x+y+z=0$$

(1) $$xyz=-6$$ --> infinitely many combinations of $$x$$, $$y$$ and $$z$$ are possible which will give different values of the expression in the stem: try x=y=1 and y=-6 or x=1, y=2, z=-3. Not sufficient.

(2) $$x+y+z=0$$ --> $$x=-(y+z)$$ --> substitute this value of x into the expression in the stem --> $$\frac{x^3+y^3+z^3}{xyz}=\frac{-(y+z)^3+y^3+z^3}{xyz}=\frac{-y^3-3y^2z-3yz^2-z^3+y^3+z^3}{xyz}=\frac{-3y^2z-3yz^2}{xyz}=\frac{-3yz(y+z)}{xyz}$$, as $$x=-(y+z)$$ then: $$\frac{-3yz(y+z)}{xyz}=\frac{-3yz*(-x)}{xyz}=\frac{3xyz}{xyz}=3$$. Sufficient.

Must know for the GMAT: $$(x+y)^3=(x+y)(x^2+2xy+y^2)=x^3+3x^2y+3xy^2+y^3$$ and $$(x-y)^3=(x-y)(x^2-2xy+y^2)=x^3-3x^2y+3xy^2-y^3$$.

_________________
Math Expert
Joined: 02 Sep 2009
Posts: 36618
Followers: 7100

Kudos [?]: 93579 [9] , given: 10578

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

### Show Tags

14 Jan 2012, 14:54
9
KUDOS
Expert's post
5
This post was
BOOKMARKED
10. If $$x$$ and $$y$$ are non-negative integers and $$x+y>0$$ is $$(x+y)^{xy}$$ an even integer?
(1) $$2^{x-y}=\sqrt[(x+y)]{16}$$
(2) $$2^x+3^y=\sqrt[(x+y)]{25}$$

(1) $$2^{x-y}=\sqrt[(x+y)]{16}$$ --> $$2^{x-y}=16^{\frac{1}{x+y}}=2^{\frac{4}{x+y}}$$ --> equate the powers: $$x-y=\frac{4}{x+y}$$ --> $$(x-y)(x+y)=4$$.

Since both $$x$$ and $$y$$ are integers (and $$x+y>0$$) then $$x-y=2$$ and $$x+y=2$$ --> $$x=2$$ and $$y=0$$ --> $$(x+y)^{xy}=2^0=1=odd$$, so the answer to the question is No. Sufficient. (Note that $$x-y=1$$ and $$x+y=4$$ --> $$x=2.5$$ and $$y=1.5$$ is not a valid scenario (solution) as both unknowns must be integers)

(2) $$2^x+3^y=\sqrt[(x+y)]{25}$$ --> obviously $$\sqrt[(x+y)]{25}$$ must be an integer (since $$2^x+3^y=integer$$) and as $$x+y=integer$$ then the only solution is $$\sqrt[(x+y)]{25}=\sqrt[2]{25}=5$$ --> $$x+y=2$$. So, $$2^x+3^y=5$$ --> two scenarios are possible:
A. $$x=2$$ and $$y=0$$ (notice that $$x+y=2$$ holds true) --> $$2^x+3^y=2^2+3^0=5$$, and in this case: $$(x+y)^{xy}=2^0=1=odd$$;
B. $$x=1$$ and $$y=1$$ (notice that $$x+y=2$$ holds true) --> $$2^x+3^y=2^1+3^1=5$$, and in this case: $$(x+y)^{xy}=2^1=2=even$$.

_________________
Math Expert
Joined: 02 Sep 2009
Posts: 36618
Followers: 7100

Kudos [?]: 93579 [4] , given: 10578

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

### Show Tags

14 Jan 2012, 14:56
4
KUDOS
Expert's post
12
This post was
BOOKMARKED
11. What is the value of $$xy$$?
(1) $$3^x*5^y=75$$
(2) $$3^{(x-1)(y-2)}=1$$

Notice that we are not told that the $$x$$ and $$y$$ are integers.

(1) $$3^x*5^y=75$$ --> if $$x$$ and $$y$$ are integers then as $$75=3^1*5^2$$ then $$x=1$$ and $$y=2$$ BUT if they are not, then for any value of $$x$$ there will exist some non-integer $$y$$ to satisfy given expression and vise-versa (for example if $$y=1$$ then $$3^x*5^y=3^x*5=75$$ --> $$3^x=25$$ --> $$x=some \ irrational \ #\approx{2.9}$$). Not sufficient.

(2) $$5^{(x-1)(y-2)}=1$$ --> $$(x-1)(y-2)=0$$ --> either $$x=1$$ and $$y$$ is ANY number (including 2) or $$y=2$$ and $$x$$ is ANY number (including 1). Not sufficient.

(1)+(2) If from (2) $$x=1$$ then from (1) $$3^x*5^y=3*5^y=75$$ --> $$y=2$$ and if from (2) $$y=2$$ then from (1) $$3^x*5^y=3^x*25=75$$ --> $$x=1$$. Thus $$x=1$$ and $$y=2$$. Sufficient.

_________________
Math Expert
Joined: 02 Sep 2009
Posts: 36618
Followers: 7100

Kudos [?]: 93579 [2] , given: 10578

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

### Show Tags

14 Jan 2012, 15:04
2
KUDOS
Expert's post
Just posted solutions. Kudos points given to everyone with correct solutions. Let me know if I missed someone.
_________________
GMAT Tutor
Joined: 24 Jun 2008
Posts: 1183
Followers: 422

Kudos [?]: 1510 [0], given: 4

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

### Show Tags

14 Jan 2012, 18:16
Bunuel wrote:
2. If x, y, and z are positive integers and $$xyz=2,700$$. Is $$\sqrt{x}$$ and integer?
(1) $$y$$ is an even perfect square and $$z$$ is an odd perfect cube.
(2) $$\sqrt{z}$$ is not an integer.

Note: a perfect square, is an integer that can be written as the square of some other integer. For example 16=4^2, is a perfect square. Similarly a perfect cube, is an integer that can be written as the cube of some other integer. For example 27=3^3, is a perfect cube.

Make prime factorization of 2,700 --> $$xyz=2^2*3^3*5^2$$.

(1) $$y$$ is an even perfect square and $$z$$ is an odd perfect cube --> if y is either 2^2 or 2^2*5^2 and $$z=3^3$$ =odd perfect square then $$x$$ must be a perfect square which makes $$\sqrt{x}$$ an integer: $$x=5^2$$ or $$x=1$$. But if $$z=1^3$$ =odd perfect square then $$x$$ could be $$3^3$$ which makes $$\sqrt{x}$$ not an integer. Not sufficient.

(2) $$\sqrt{z}$$ is not an integer. Clearly insufficient.

(1)+(2) As from (1) $$\sqrt{z}\neq{integer}$$ then $$z\neq{1}$$, therefore it must be $$3^3$$ (from 1) --> $$x$$ is a perfect square which makes $$\sqrt{x}$$ an integer: $$x=5^2$$ or $$x=1$$. Sufficient.

In the highlighted part above, a couple of possibilities are missing. If y is an even perfect square, then we know y is divisible by 2^2. It may or may not be divisible by 3^2 and 5^2 (two choices each way, so four possibilities). So y could have any of four values:

2^2
(2^2)(3^2)
(2^2)(5^2)
(2^2)(3^2)(5^2)

Each of these values is possible when z = 1. While it doesn't affect the answer, it is also possible for x to be equal to 3 here, if y is equal to (2^2)(3^2)(5^2) and z is equal to 1, and x can also be (3)(5^2). I think you also meant to write 'odd perfect cube' instead of 'odd perfect square' (blue highlights).
_________________

GMAT Tutor in Toronto

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

Math Expert
Joined: 02 Sep 2009
Posts: 36618
Followers: 7100

Kudos [?]: 93579 [0], given: 10578

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

### Show Tags

15 Jan 2012, 01:39
IanStewart wrote:
Bunuel wrote:
2. If x, y, and z are positive integers and $$xyz=2,700$$. Is $$\sqrt{x}$$ and integer?
(1) $$y$$ is an even perfect square and $$z$$ is an odd perfect cube.
(2) $$\sqrt{z}$$ is not an integer.

Note: a perfect square, is an integer that can be written as the square of some other integer. For example 16=4^2, is a perfect square. Similarly a perfect cube, is an integer that can be written as the cube of some other integer. For example 27=3^3, is a perfect cube.

Make prime factorization of 2,700 --> $$xyz=2^2*3^3*5^2$$.

(1) $$y$$ is an even perfect square and $$z$$ is an odd perfect cube --> if y is either 2^2 or 2^2*5^2 and $$z=3^3$$ =odd perfect square then $$x$$ must be a perfect square which makes $$\sqrt{x}$$ an integer: $$x=5^2$$ or $$x=1$$. But if $$z=1^3$$ =odd perfect square then $$x$$ could be $$3^3$$ which makes $$\sqrt{x}$$ not an integer. Not sufficient.

(2) $$\sqrt{z}$$ is not an integer. Clearly insufficient.

(1)+(2) As from (1) $$\sqrt{z}\neq{integer}$$ then $$z\neq{1}$$, therefore it must be $$3^3$$ (from 1) --> $$x$$ is a perfect square which makes $$\sqrt{x}$$ an integer: $$x=5^2$$ or $$x=1$$. Sufficient.

In the highlighted part above, a couple of possibilities are missing. If y is an even perfect square, then we know y is divisible by 2^2. It may or may not be divisible by 3^2 and 5^2 (two choices each way, so four possibilities). So y could have any of four values:

2^2
(2^2)(3^2)
(2^2)(5^2)
(2^2)(3^2)(5^2)

Each of these values is possible when z = 1. While it doesn't affect the answer, it is also possible for x to be equal to 3 here, if y is equal to (2^2)(3^2)(5^2) and z is equal to 1, and x can also be (3)(5^2). I think you also meant to write 'odd perfect cube' instead of 'odd perfect square' (blue highlights).

Yes, I know Ian. For (1) I just discussed two possible scenarios to get an YES and NO answers to discard this statement, rather than listing all possibilities as you did. Thank you though for elaborating more on other cases.
_________________
Manager
Joined: 18 May 2011
Posts: 71
Followers: 0

Kudos [?]: 3 [0], given: 179

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

### Show Tags

17 Jan 2012, 10:44
Bunuel wrote:
7. If $$x$$ is a positive integer is $$\sqrt{x}$$ an integer?
(1) $$\sqrt{7*y}$$ is an integer
(2) $$\sqrt{9*x}$$ is not an integer

Must know for the GMAT: if $$x$$ is a positive integer then $$\sqrt{x}$$ is either a positive integer itself or an irrational number. (It can not be some reduced fraction eg 7/3 or 1/2)

Also note that the question basically asks whether $$x$$ is a perfect square.

(1) $$\sqrt{7*x}$$ is an integer --> $$x$$ can not be a perfect square because if it is, for example if $$x=n^2$$ for some positive integer $$n$$ then $$\sqrt{7x}=\sqrt{7n^2}=n\sqrt{7}\neq{integer}$$. Sufficient.

(2) $$\sqrt{9*x}$$ is not an integer --> $$\sqrt{9*x}=3*\sqrt{x}\neq{integer}$$ --> $$\sqrt{x}\neq{integer}$$. Sufficient.

Originally I chose B because there was no mentioned of Y in the original answer choice until I read the rest of your explanation and realized it was probably just a typo.

By the way thank you for compiling such a nice collection of Math problems Bunuel!
Manager
Joined: 25 Aug 2011
Posts: 193
Location: India
GMAT 1: 730 Q49 V40
WE: Operations (Insurance)
Followers: 1

Kudos [?]: 292 [0], given: 11

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

### Show Tags

17 Mar 2012, 05:28
Bunuel cant sqrt[x] be negative eg .. sqrt[4] = + or - 2.. in that case how is a suffucuent

SonyGmat wrote:
Bunuel wrote:
4. If $$xyz\neq{0}$$ is $$(x^{-4})*(\sqrt[3]{y})*(z^{-2})<0$$?
(1) $$\sqrt[5]{y}>\sqrt[4]{x^2}$$
(2) $$y^3>\frac{1}{z{^4}}$$

if we can find the sign of y we can answer the question.

(1) $$\sqrt[5]{y}>\sqrt[4]{x^2}>0$$ since it is$$[\sqrt[4]{x^2}]=[\sqrt[]{x}]$$

Sufficient

(2) $$y^3>\frac{1}{z{^4}}>0$$ since $$z^4>0$$

Sufficient

Therefore, D
Math Expert
Joined: 02 Sep 2009
Posts: 36618
Followers: 7100

Kudos [?]: 93579 [1] , given: 10578

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

### Show Tags

17 Mar 2012, 05:33
1
KUDOS
Expert's post
1
This post was
BOOKMARKED
devinawilliam83 wrote:
Bunuel cant sqrt[x] be negative eg .. sqrt[4] = + or - 2.. in that case how is a suffucuent

1. GMAT is dealing only with Real Numbers: Integers, Fractions and Irrational Numbers.

2. Any nonnegative real number has a unique non-negative square root called the principal square root and unless otherwise specified, the square root is generally taken to mean the principal square root.

When the GMAT provides the square root sign for an even root, such as $$\sqrt{x}$$ or $$\sqrt[4]{x}$$, then the only accepted answer is the positive root.

That is, $$\sqrt{25}=5$$, NOT +5 or -5. In contrast, the equation $$x^2=25$$ has TWO solutions, $$\sqrt{25}=+5$$ and $$-\sqrt{25}=-5$$.

So, remember: even roots have only non-negative value on the GMAT.

Hope it's clear.
_________________
Senior Manager
Joined: 12 Dec 2010
Posts: 282
Concentration: Strategy, General Management
GMAT 1: 680 Q49 V34
GMAT 2: 730 Q49 V41
GPA: 4
WE: Consulting (Other)
Followers: 9

Kudos [?]: 46 [0], given: 23

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

### Show Tags

06 Apr 2012, 23:40
Bunuel wrote:
9. If $$x$$, $$y$$ and $$z$$ are non-zero numbers, what is the value of $$\frac{x^3+y^3+z^3}{xyz}$$?
(1) $$xyz=-6$$
(2) $$x+y+z=0$$

(1) $$xyz=-6$$ --> infinitely many combinations of $$x$$, $$y$$ and $$z$$ are possible which will give different values of the expression in the stem: try x=y=1 and y=-6 or x=1, y=2, z=-3. Not sufficient.

(2) $$x+y+z=0$$ --> $$x=-(y+z)$$ --> substitute this value of x into the expression in the stem --> $$\frac{x^3+y^3+z^3}{xyz}=\frac{-(y+z)^3+y^3+z^3}{xyz}=\frac{-y^3-3y^2z-3yz^2-z^3+y^3+z^3}{xyz}=\frac{-3y^2z-3yz^2}{xyz}=\frac{-3yz(y+z)}{xyz}$$, as $$x=-(y+z)$$ then: $$\frac{-3yz(y+z)}{xyz}=\frac{-3yz*(-x)}{xyz}=\frac{3xyz}{xyz}=3$$. Sufficient.

Must know for the GMAT: $$(x+y)^3=(x+y)(x^2+2xy+y^2)=x^3+3x^2y+3xy^2+y^3$$ and $$(x-y)^3=(x-y)(x^2-2xy+y^2)=x^3-3x^2y+3xy^2-y^3$$.

Actually there is a direct formula for cubes of three entities-

$$x^3+y^3+z^3 - 3*x*y*z = (x+y+z) ( x^2+y^2+z^2 - x*y - y*z - z*x)$$
so if one can figure out either of the quantity on RHS is 0 then you have the answer for the Q given...
_________________

My GMAT Journey 540->680->730!

~ When the going gets tough, the Tough gets going!

Moderator
Joined: 01 Sep 2010
Posts: 3097
Followers: 787

Kudos [?]: 6565 [0], given: 1014

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

### Show Tags

13 Apr 2012, 18:14
Bunuel wrote:
2. If x, y, and z are positive integers and $$xyz=2,700$$. Is $$\sqrt{x}$$ and integer?
(1) $$y$$ is an even perfect square and $$z$$ is an odd perfect cube.
(2) $$\sqrt{z}$$ is not an integer.

Note: a perfect square, is an integer that can be written as the square of some other integer. For example 16=4^2, is a perfect square. Similarly a perfect cube, is an integer that can be written as the cube of some other integer. For example 27=3^3, is a perfect cube.

Make prime factorization of 2,700 --> $$xyz=2^2*3^3*5^2$$.

(1) $$y$$ is an even perfect square and $$z$$ is an odd perfect cube --> if $$y$$ is either $$2^2$$ or $$2^2*5^2$$ and $$z=3^3=odd \ perfect \ square$$ then $$x$$ must be a perfect square which makes $$\sqrt{x}$$ an integer: $$x=5^2$$ or $$x=1$$. But if $$z=1^3=odd \ perfect \ cube$$ then $$x$$ could be $$3^3$$ which makes $$\sqrt{x}$$ not an integer. Not sufficient.

(2) $$\sqrt{z}$$ is not an integer. Clearly insufficient.

(1)+(2) As from (1) $$\sqrt{z}\neq{integer}$$ then $$z\neq{1}$$, therefore it must be $$3^3$$ (from 1) --> $$x$$ is a perfect square which makes $$\sqrt{x}$$ an integer: $$x=5^2$$ or $$x=1$$. Sufficient.

Bunuel can you please elaborate from where we have these values. I missed the link . x=1 and z=1^3

Thanks
_________________
Math Expert
Joined: 02 Sep 2009
Posts: 36618
Followers: 7100

Kudos [?]: 93579 [0], given: 10578

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

### Show Tags

14 Apr 2012, 11:30
carcass wrote:
Bunuel wrote:
2. If x, y, and z are positive integers and $$xyz=2,700$$. Is $$\sqrt{x}$$ and integer?
(1) $$y$$ is an even perfect square and $$z$$ is an odd perfect cube.
(2) $$\sqrt{z}$$ is not an integer.

Note: a perfect square, is an integer that can be written as the square of some other integer. For example 16=4^2, is a perfect square. Similarly a perfect cube, is an integer that can be written as the cube of some other integer. For example 27=3^3, is a perfect cube.

Make prime factorization of 2,700 --> $$xyz=2^2*3^3*5^2$$.

(1) $$y$$ is an even perfect square and $$z$$ is an odd perfect cube --> if $$y$$ is either $$2^2$$ or $$2^2*5^2$$ and $$z=3^3=odd \ perfect \ square$$ then $$x$$ must be a perfect square which makes $$\sqrt{x}$$ an integer: $$x=5^2$$ or $$x=1$$. But if $$z=1^3=odd \ perfect \ cube$$ then $$x$$ could be $$3^3$$ which makes $$\sqrt{x}$$ not an integer. Not sufficient.

(2) $$\sqrt{z}$$ is not an integer. Clearly insufficient.

(1)+(2) As from (1) $$\sqrt{z}\neq{integer}$$ then $$z\neq{1}$$, therefore it must be $$3^3$$ (from 1) --> $$x$$ is a perfect square which makes $$\sqrt{x}$$ an integer: $$x=5^2$$ or $$x=1$$. Sufficient.

Bunuel can you please elaborate from where we have these values. I missed the link . x=1 and z=1^3

Thanks

Are you talking about statement 1? The values of x, y, and z analyzed there are just possible values that satisfy statement 1.

Given that: $$xyz=2^2*3^3*5^2$$ and (1) says that $$y$$ is an even perfect square and $$z$$ is an odd perfect cube.

Now, if for example $$y=2^2*5^2=even \ perfect \ square$$ and $$z=1^3=odd \ perfect \ cube$$ then $$x=3^3$$. You can apply the similar logic to obtain other possible values of x, y, and z.
_________________
Manager
Joined: 06 Apr 2012
Posts: 51
Followers: 1

Kudos [?]: 10 [0], given: 25

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

### Show Tags

27 Apr 2012, 10:46
Bunuel wrote:
devinawilliam83 wrote:
Bunuel cant sqrt[x] be negative eg .. sqrt[4] = + or - 2.. in that case how is a suffucuent

1. GMAT is dealing only with Real Numbers: Integers, Fractions and Irrational Numbers.

2. Any nonnegative real number has a unique non-negative square root called the principal square root and unless otherwise specified, the square root is generally taken to mean the principal square root.

When the GMAT provides the square root sign for an even root, such as $$\sqrt{x}$$ or $$\sqrt[4]{x}$$, then the only accepted answer is the positive root.

That is, $$\sqrt{25}=5$$, NOT +5 or -5. In contrast, the equation $$x^2=25$$ has TWO solutions, $$\sqrt{25}=+5$$ and $$-\sqrt{25}=-5$$.

So, remember: even roots have only non-negative value on the GMAT.

Hope it's clear.

but won't this leave one in a dilemma?
do i take this info with a pinch of salt, or is gmat only interested in the principal square root?
Math Expert
Joined: 02 Sep 2009
Posts: 36618
Followers: 7100

Kudos [?]: 93579 [0], given: 10578

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

### Show Tags

27 Apr 2012, 10:48
nrmlvrm wrote:
Bunuel wrote:
devinawilliam83 wrote:
Bunuel cant sqrt[x] be negative eg .. sqrt[4] = + or - 2.. in that case how is a suffucuent

1. GMAT is dealing only with Real Numbers: Integers, Fractions and Irrational Numbers.

2. Any nonnegative real number has a unique non-negative square root called the principal square root and unless otherwise specified, the square root is generally taken to mean the principal square root.

When the GMAT provides the square root sign for an even root, such as $$\sqrt{x}$$ or $$\sqrt[4]{x}$$, then the only accepted answer is the positive root.

That is, $$\sqrt{25}=5$$, NOT +5 or -5. In contrast, the equation $$x^2=25$$ has TWO solutions, $$\sqrt{25}=+5$$ and $$-\sqrt{25}=-5$$.

So, remember: even roots have only non-negative value on the GMAT.

Hope it's clear.

but won't this leave one in a dilemma?
do i take this info with a pinch of salt, or is gmat only interested in the principal square root?

No ambiguity there whatsoever: even roots have only non-negative value on the GMAT Again: $$\sqrt{25}=5$$, NOT +5 or -5.
_________________
Manager
Joined: 28 Dec 2012
Posts: 115
Location: India
Concentration: Strategy, Finance
GMAT 1: Q V
WE: Engineering (Energy and Utilities)
Followers: 3

Kudos [?]: 66 [0], given: 90

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

### Show Tags

12 Jan 2013, 11:06
Bunuel wrote:
2. If x, y, and z are positive integers and $$xyz=2,700$$. Is $$\sqrt{x}$$ and integer?
(1) $$y$$ is an even perfect square and $$z$$ is an odd perfect cube.
(2) $$\sqrt{z}$$ is not an integer.

Note: a perfect square, is an integer that can be written as the square of some other integer. For example 16=4^2, is a perfect square. Similarly a perfect cube, is an integer that can be written as the cube of some other integer. For example 27=3^3, is a perfect cube.

Make prime factorization of 2,700 --> $$xyz=2^2*3^3*5^2$$.

(1) $$y$$ is an even perfect square and $$z$$ is an odd perfect cube --> if $$y$$ is either $$2^2$$ or $$2^2*5^2$$ and $$z=3^3=odd \ perfect \ square$$ then $$x$$ must be a perfect square which makes $$\sqrt{x}$$ an integer: $$x=5^2$$ or $$x=1$$. But if $$z=1^3=odd \ perfect \ cube$$ then $$x$$ could be $$3^3$$ which makes $$\sqrt{x}$$ not an integer. Not sufficient.

(2) $$\sqrt{z}$$ is not an integer. Clearly insufficient.

(1)+(2) As from (1) $$\sqrt{z}\neq{integer}$$ then $$z\neq{1}$$, therefore it must be $$3^3$$ (from 1) --> $$x$$ is a perfect square which makes $$\sqrt{x}$$ an integer: $$x=5^2$$ or $$x=1$$. Sufficient.

Hi..

Why Not B?

If root of Z is not integer, then z must be 3^3 and this make root x an integer......
Please let me know where I am wrong...

thanks.
_________________

Impossibility is a relative concept!!

Math Expert
Joined: 02 Sep 2009
Posts: 36618
Followers: 7100

Kudos [?]: 93579 [0], given: 10578

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

### Show Tags

13 Jan 2013, 02:39
sagarsingh wrote:
Bunuel wrote:
2. If x, y, and z are positive integers and $$xyz=2,700$$. Is $$\sqrt{x}$$ and integer?
(1) $$y$$ is an even perfect square and $$z$$ is an odd perfect cube.
(2) $$\sqrt{z}$$ is not an integer.

Note: a perfect square, is an integer that can be written as the square of some other integer. For example 16=4^2, is a perfect square. Similarly a perfect cube, is an integer that can be written as the cube of some other integer. For example 27=3^3, is a perfect cube.

Make prime factorization of 2,700 --> $$xyz=2^2*3^3*5^2$$.

(1) $$y$$ is an even perfect square and $$z$$ is an odd perfect cube --> if $$y$$ is either $$2^2$$ or $$2^2*5^2$$ and $$z=3^3=odd \ perfect \ square$$ then $$x$$ must be a perfect square which makes $$\sqrt{x}$$ an integer: $$x=5^2$$ or $$x=1$$. But if $$z=1^3=odd \ perfect \ cube$$ then $$x$$ could be $$3^3$$ which makes $$\sqrt{x}$$ not an integer. Not sufficient.

(2) $$\sqrt{z}$$ is not an integer. Clearly insufficient.

(1)+(2) As from (1) $$\sqrt{z}\neq{integer}$$ then $$z\neq{1}$$, therefore it must be $$3^3$$ (from 1) --> $$x$$ is a perfect square which makes $$\sqrt{x}$$ an integer: $$x=5^2$$ or $$x=1$$. Sufficient.

Hi..

Why Not B?

If root of Z is not integer, then z must be 3^3 and this make root x an integer......
Please let me know where I am wrong...

thanks.

There are other cases possible. For example, z=3, y=2*3*5^2 and x=2.

Hope it's clear.
_________________
Senior Manager
Joined: 07 Sep 2010
Posts: 336
Followers: 6

Kudos [?]: 665 [0], given: 136

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

### Show Tags

03 Jun 2013, 06:58
Bunuel wrote:
7. If $$x$$ is a positive integer is $$\sqrt{x}$$ an integer?
(1) $$\sqrt{7*x}$$ is an integer
(2) $$\sqrt{9*x}$$ is not an integer

Hi Bunuel,
I would like you to point my mistake in evaluating equation 1

Lets Say:
$$\sqrt{7*x}$$ = $$K$$(an integer)
Squaring both sides
$$7*x$$= $$K^2$$
$$x = (K^2)/7$$
Now, Since X is a positive integer, this means that K^2 has to be a multiple of 7.
If K^2 = 7 then, x = 1 and $$\sqrt{x}$$= Integer.
But,as per your explanation, x can't be an integer.

Regards,
imhimanshu
_________________

+1 Kudos me, Help me unlocking GMAT Club Tests

Math Expert
Joined: 02 Sep 2009
Posts: 36618
Followers: 7100

Kudos [?]: 93579 [0], given: 10578

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

### Show Tags

03 Jun 2013, 07:01
imhimanshu wrote:
Bunuel wrote:
7. If $$x$$ is a positive integer is $$\sqrt{x}$$ an integer?
(1) $$\sqrt{7*x}$$ is an integer
(2) $$\sqrt{9*x}$$ is not an integer

Hi Bunuel,
I would like you to point my mistake in evaluating equation 1

Lets Say:
$$\sqrt{7*x}$$ = $$K$$(an integer)
Squaring both sides
$$7*x$$= $$K^2$$
$$x = (K^2)/7$$
Now, Since X is a positive integer, this means that K^2 has to be a multiple of 7.
If K^2 = 7 then, x = 1 and $$\sqrt{x}$$= Integer.
But,as per your explanation, x can't be an integer.

Regards,
imhimanshu

If k^2=7, then k won't be an integer as we defined.

Hope it helps.
_________________
Re: NEW!!! Tough and tricky exponents and roots questions   [#permalink] 03 Jun 2013, 07:01

Go to page   Previous    1   2   3   4   5    Next  [ 94 posts ]

Similar topics Replies Last post
Similar
Topics:
If the square root of t is a real number, is the square root 3 09 Apr 2013, 15:32
16 Is root{x} a prime number? 23 26 Jan 2012, 05:41
6 Factoring w/ Exponents Question 9 06 Jan 2011, 12:37
23 Series A(n) is such that i*A(i) = j*A(j) for any pair of 19 30 Jun 2010, 09:38
1 Is it Ok to take the roots first?For example, in question 1 5 28 Aug 2010, 12:49
Display posts from previous: Sort by