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# If z^n = 1, what is the value of z?

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If z^n = 1, what is the value of z?  [#permalink]

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24 Jan 2011, 08:02
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55% (hard)

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57% (01:14) correct 43% (01:15) wrong based on 748 sessions

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If z^n = 1, what is the value of z?

(1) n is a non zero integer
(2) z > 0
Math Expert
Joined: 02 Sep 2009
Posts: 59125
Re: If z^n = 1, what is the value of z?  [#permalink]

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29 Jun 2012, 05:59
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Stiv wrote:
If z^n = 1, what is the value of z?

(1) n is a nonzero integer.
(2) z > 0

Zero raised to any power is zero and any number raised to the power of 0 equals one?
Is that the rule of it is reversed?

I'm not sure understand the red part in your post above.

If z^n = 1, what is the value of z?

(1) n is a non zero integer --> $$1^{any \ integer}=1$$ and also $$(-1)^{even}=1$$, so $$z$$ can be 1 or -1. Not sufficient.

(2) z > 0 --> any nonzero number to the power of 0 is 1, so if $$n=0$$ then $$z$$ can be any non-zero number (any positive number in our case as given that $$z>0$$). Not sufficient.

(1)+(2) $$n$$ is a nonzero integer and $$z>0$$ implies that $$z$$ can equal to 1 only. Sufficient.

For more on number theory and exponents check: math-number-theory-88376.html

DS questions on exponents: search.php?search_id=tag&tag_id=39
PS questions on exponents: search.php?search_id=tag&tag_id=60

Tough and tricky DS exponents and roots questions with detailed solutions: tough-and-tricky-exponents-and-roots-questions-125967.html
Tough and tricky PS exponents and roots questions with detailed solutions: tough-and-tricky-exponents-and-roots-questions-125956.html

Hope it helps.
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Re: If z^n = 1, what is the value of z?  [#permalink]

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27 Dec 2011, 11:03
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Statement 1 is insufficient because Z could be 1 or -1. Let's say n is 2. 1^2 = 1 and (-1)^2 = 1

Statement 2 is insufficient because if n = 0, then Z could be one of many numbers, such as 1 or 2. If Z = 1, 1^0 = 1. If Z = 2, 2^0 = 1. Any non-zero number raised to 0 equals 1.

Statements 1 and 2 combined are sufficient because with statement 1, we know Z is either 1 or -1 and with statement 2 we know Z is greater than 0. So Z must be 1.
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Re: Gmat Prep Number Properties  [#permalink]

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24 Jan 2011, 08:10
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(1) n is a non-zero integer implies n can be positive or negative integer, which is NOT SUFFICIENT to find z.

(2) z > 0 implies that z is any positive value, which is again NOT SUFFICIENT.

Combining (1) and (2), we have n is a non-zero integer and z > 0. So, the only possible value of z = 1, and n can be any positive integer, say, 1, 2, 3, 4....
Hence, combining the statements the question can be answered.

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Re: Gmat Prep Number Properties  [#permalink]

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24 Jan 2011, 08:16
I guess my question is , why cant it be B by itself ? Because for the equation to equal 1, what other value can z have knowing its positive, and in that case what would n equal to?

Can you give an example of b insufficient

Thanks
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Re: Gmat Prep Number Properties  [#permalink]

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24 Jan 2011, 08:21
I guess my question is , why cant it be B by itself ? Because for the equation to equal 1, what other value can z have knowing its positive, and in that case what would n equal to?

Can you give an example of b insufficient

Thanks

Since z > 0, so z can be 2, 3, 4, 5, 6...or even any fractional value say, $$\frac{1}{2}$$, $$\frac{3}{4}$$...
Now if we take n = 0, then $$2^0 = 1$$
$$3^0 = 1$$
Similarly $$(\frac{1}{2})^0 = 1$$
We don't have any unique value of z. So, statement 2 alone is NOT SUFFICIENT.
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Re: If z^n = 1, what is the value of z?  [#permalink]

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29 Jun 2012, 05:50
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If z^n = 1, what is the value of z?

(1) n is a nonzero integer.
(2) z > 0

Zero raised to any power is zero and any number raised to the power of 0 equals one?
Is that the rule of it is reversed?
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Re: If z^n = 1, what is the value of z?  [#permalink]

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08 Jan 2015, 05:48
2
1
Stiv wrote:
If z^n = 1, what is the value of z?

(1) n is a nonzero integer.
(2) z > 0

Zero raised to any power is zero and any number raised to the power of 0 equals one?
Is that the rule of it is reversed?

This is not a difficult question by itself, but clarity of approach matters a lot in getting it right. The low accuracy for this question suggests that most students could not think through this question clearly.

At eGMAT, we strongly advocate that in DS Questions, the student should first analyze the question statement thoroughly and only then move on to analyzing the two statements. You'll see how elegantly this question will simplify with this approach.

We are given that z^n = 1. So, what cases are possible for the value of z and n?

Case 1: z = 1; n has any integral value
Case 2: z = -1; n is an even integer
Case 3: z has any non-zero value; n = 0

Please note that only after this analysis are we going to the first Statement.

As per the first statement,
n is a non-zero integer
This rules out Case 3.
However, this still leaves out Case 1 and 2. So, z can either be equal to 1 or z can be equal to -1. So, Statement 1 alone is not sufficient.

As per the second statement,
z > 0
This rules out Case 2. However, Case 1 and 3 still remain. Again, we have not been able to determine a unique value of z. So, Statement 2 alone is not sufficient either.

Combining both the Statements,
From Statement 1, z could either be 1 or -1
From Statement 2, z > 0
Therefore, only possible value of z is 1.

Thus, by combining both the statements together, we have been able to determine a unique value of z. So, the correct answer is Choice C.

Takeaway: The correct answer is only a byproduct of a clear approach.

Hope this helps.

Japinder
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Re: If z^n = 1, what is the value of z?  [#permalink]

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21 Mar 2015, 22:11
Bunuel wrote:
(1) n is a non zero integer --> $$1^{any \ integer}=1$$ and also $$(-1)^{even}=1$$, so $$z$$ can be 1 or -1. Not sufficient.

(2) z > 0 --> any nonzero number to the power of 0 is 1, so if $$n=0$$ then $$z$$ can be any non-zero number (any positive number in our case as given that $$z>0$$). Not sufficient.

(1)+(2) $$n$$ is a nonzero integer and $$z>0$$ implies that $$z$$ can equal to 1 only. Sufficient.

Can we take $$(\sqrt{1})^2$$ as a plug in value to check these statements..
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Re: If z^n = 1, what is the value of z?  [#permalink]

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22 Mar 2015, 06:13
1
suhasancd wrote:
Bunuel wrote:
(1) n is a non zero integer --> $$1^{any \ integer}=1$$ and also $$(-1)^{even}=1$$, so $$z$$ can be 1 or -1. Not sufficient.

(2) z > 0 --> any nonzero number to the power of 0 is 1, so if $$n=0$$ then $$z$$ can be any non-zero number (any positive number in our case as given that $$z>0$$). Not sufficient.

(1)+(2) $$n$$ is a nonzero integer and $$z>0$$ implies that $$z$$ can equal to 1 only. Sufficient.

Can we take $$(\sqrt{1})^2$$ as a plug in value to check these statements..

$$\sqrt{1}=1$$, so are you asking whether we can plug 1 for n? Well, yes we can...
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Re: If z^n = 1, what is the value of z?  [#permalink]

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22 Mar 2015, 06:23
Bunuel wrote:

$$\sqrt{1}=1$$, so are you asking whether we can plug 1 for n? Well, yes we can...

Oh.. ya.. I forgot that the square root of a perfect number is always +ve.

$$\sqrt{36}$$ = 6 (not -6)

So $$\sqrt{1}$$ = 1 (not -1)

I had +/- 1 in my head while combining statements I and II together.

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Re: If z^n = 1, what is the value of z?  [#permalink]

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22 Mar 2015, 06:24
suhasancd wrote:
Bunuel wrote:

$$\sqrt{1}=1$$, so are you asking whether we can plug 1 for n? Well, yes we can...

Oh.. ya.. I forgot that the square root of a perfect number is always +ve.

$$\sqrt{36}$$ = 6 (not -6)

So $$\sqrt{1}$$ = 1 (not -1)

I had +/- 1 in my head while combining statements I and II together.

Any even root from any postie number is positive.
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Re: If z^n = 1, what is the value of z?  [#permalink]

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09 Sep 2015, 18:19
Hi All,

This exponent rule 'concept' is something of a classic in the realm of standardized testing - it serves as a relatively simple way to assess a Test Taker's 'thoroughness of understanding' on a specific concept:

Here, the concept is "using exponent rules, and one number raised to one exponent, how many different ways can you get to the number 1?"

The first ('obvious') answer is "1 raised to any power = 1"

eg. 1^2, 1^50, 1^(-3), etc.

There are OTHER possibilities though. If your base is -1, then any EVEN exponent will lead us to a total of 1...

eg. (-1)^2, (-1)^4, (-1)^(-2), etc.

Finally, raising any number to the '0 power' will also give us a total of 1...

eg. 1^0, 537^0, (-13)^0, etc.

When dealing with this specific situation, it's important to pay careful attention to the information that you're given. What do you really know about the 'base' and the 'power' involved? If you don't know anything, then you have to consider all of the above possibilities.

GMAT assassins aren't born, they're made,
Rich
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Re: If z^n = 1, what is the value of z?  [#permalink]

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16 Jan 2019, 03:07
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