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# If x is an integer, is 9^x + 9^{-x} = b ?

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If x is an integer, is 9^x + 9^{-x} = b ? [#permalink]

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01 Oct 2012, 05:20
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The Official Guide for GMAT® Review, 13th Edition - Quantitative Questions Project

If x is an integer, is $$9^x + 9^{-x} = b$$ ?

(1) $$3^x + 3^{-x} = \sqrt{b + 2}$$
(2) x > 0

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Re: If x is an integer, is 9^x + 9^{-x} = b ? [#permalink]

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01 Oct 2012, 05:20
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SOLUTION

If x is an integer, is $$9^x + 9^{-x} = b$$ ?

(1) $$3^x + 3^{-x} = \sqrt{b + 2}$$ --> square both sides --> $$9^x+2*3^x*\frac{1}{3^x}+9^{-x}=b+2$$ --> $$9^x + 9^{-x} = b$$. So answer to the question is YES. Sufficient.

(2) x > 0. No sufficient.

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Re: If x is an integer, is 9^x + 9^{-x} = b ? [#permalink]

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01 Oct 2012, 05:40
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1) $$3^x + 3^-^x = \sqrt{b + 2}$$

square both sides

$$9^x+2*3^x*\frac{1}{3^x}+9^{-x} = b+2$$

Therefore, $$9^x + 9^{-x} = b$$.

Sufficient.

2) Tells us nothing about b, but rather that x is a positive number. Insufficient.

[Reveal] Spoiler:
Knowing this, the solution is A - statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

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Re: If x is an integer, is 9^x + 9^{-x} = b ? [#permalink]

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01 Oct 2012, 05:54
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This was tough. I spent at least 3 minutes (during the exam is not possible)

We can rephrase the stimulus but I see a lot of options to do. Is not straight (at least for me). Ok better to look at the statements

2) this say nothing about $$=$$between the left part of equation and the right part INSUFF

1) square boot sides so we have $$(3^x + 3^-x)^2$$$$=$$$$\sqrt{b + 2}^2$$

Now we 'd have $$9^x$$ that is, is the same of $$3^2x$$ -------> $$9^x + 9^-x + 2 ( 3^x + 3^-x)$$$$=$$ $$b + 2$$ ---------> $$9^x + 2 + 9^-x = b + 2$$

In the end $$9^x + 9^-x = b$$ SUFF

Note: $$2 ( 3^x + 3^-x)$$ is zero because we have $$3^ x- x$$ . a number power zero is 1 ---> $$2*1 = 2$$ for me is more than 600 level
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Re: If x is an integer, is 9^x + 9^{-x} = b ? [#permalink]

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01 Oct 2012, 20:18
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1) Square both the side & get the same equation as in stem --->Sufficient
2) Insufficient
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Re: If x is an integer, is 9^x + 9^{-x} = b ? [#permalink]

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04 Oct 2012, 14:44
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SOLUTION

If x is an integer, is $$9^x + 9^{-x} = b$$ ?

(1) $$3^x + 3^{-x} = \sqrt{b + 2}$$ --> square both sides --> $$9^x+2*3^x*\frac{1}{3^x}+9^{-x}=b+2$$ --> $$9^x + 9^{-x} = b$$. So answer to the question is YES. Sufficient.

(2) x > 0. No sufficient.

Kudos points given to everyone with correct solution. Let me know if I missed someone.
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Re: If x is an integer, is 9^x + 9^{-x} = b ? [#permalink]

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31 Oct 2012, 23:40
Bunuel wrote:
SOLUTION

If x is an integer, is $$9^x + 9^{-x} = b$$ ?

(1) $$3^x + 3^{-x} = \sqrt{b + 2}$$ --> square both sides --> $$9^x+2*3^x*\frac{1}{3^x}+9^{-x}=b+2$$ --> $$9^x + 9^{-x} = b$$. So answer to the question is YES. Sufficient.

(2) x > 0. No sufficient.

Kudos points given to everyone with correct solution. Let me know if I missed someone.

Question please: Can we do anything with the 9^x + 9^{-x} = b or simplify any more than what is given? I tried to do something more but could not find anything proper...
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Re: If x is an integer, is 9^x + 9^{-x} = b ? [#permalink]

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01 Nov 2012, 07:25
ikokurin wrote:
Bunuel wrote:
SOLUTION

If x is an integer, is $$9^x + 9^{-x} = b$$ ?

(1) $$3^x + 3^{-x} = \sqrt{b + 2}$$ --> square both sides --> $$9^x+2*3^x*\frac{1}{3^x}+9^{-x}=b+2$$ --> $$9^x + 9^{-x} = b$$. So answer to the question is YES. Sufficient.

(2) x > 0. No sufficient.

Kudos points given to everyone with correct solution. Let me know if I missed someone.

Question please: Can we do anything with the 9^x + 9^{-x} = b or simplify any more than what is given? I tried to do something more but could not find anything proper...

I'd say $$9^x + 9^{-x}$$ the simplest way of writing this expression and as you can see from the solution we don't even need to manipulate with it further to answer the question.
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Re: If x is an integer, is 9^x + 9^{-x} = b ? [#permalink]

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10 Nov 2012, 22:22
bb, I tried navigating below link, but it says "You are not authorised to read this forum". Can you check please?

the-official-guide-for-gmat-review-13th-edition-quant-134495.html
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Re: If x is an integer, is 9^x + 9^{-x} = b ? [#permalink]

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11 Nov 2012, 05:42
prashantponde wrote:
bb, I tried navigating below link, but it says "You are not authorised to read this forum". Can you check please?

the-official-guide-for-gmat-review-13th-edition-quant-134495.html

This thread is no longer available.
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Re: If x is an integer, is 9^x + 9^{-x} = b ? [#permalink]

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13 Nov 2012, 18:05
why is 3^x * 3^x = 9x? shouldnt it be 9^2x?
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Re: If x is an integer, is 9^x + 9^{-x} = b ? [#permalink]

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14 Nov 2012, 02:51
JJ2014 wrote:
why is 3^x * 3^x = 9x? shouldnt it be 9^2x?

$$3^x * 3^x=3^{x+x}=3^{2x}=9^x$$.

For more check Number Theory chapter of our Math Book: math-number-theory-88376.html

Hope it helps.
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Re: If x is an integer, is 9^x + 9^{-x} = b ? [#permalink]

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Re: If x is an integer, is 9^x + 9^{-x} = b ? [#permalink]

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17 Nov 2014, 15:24
As was noted earlier in this post, this question uses the formula (x+y)^2=x^2+2xy+y^2. In this question xy=1 because x^0=1. This allows for the 2's to cancel out.

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Re: If x is an integer, is 9^x + 9^{-x} = b ? [#permalink]

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05 May 2015, 15:02
Bunuel wrote:
SOLUTION

If x is an integer, is $$9^x + 9^{-x} = b$$ ?

(1) $$3^x + 3^{-x} = \sqrt{b + 2}$$ --> square both sides --> $$9^x+2*3^x*\frac{1}{3^x}+9^{-x}=b+2$$ --> $$9^x + 9^{-x} = b$$. So answer to the question is YES. Sufficient.

(2) x > 0. No sufficient.

When you square both sides, I don't understand where the +2 comes from? Can you please explain?
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Re: If x is an integer, is 9^x + 9^{-x} = b ? [#permalink]

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05 May 2015, 15:53
dutchmen991 wrote:
Bunuel wrote:
SOLUTION

If x is an integer, is $$9^x + 9^{-x} = b$$ ?

(1) $$3^x + 3^{-x} = \sqrt{b + 2}$$ --> square both sides --> $$9^x+2*3^x*\frac{1}{3^x}+9^{-x}=b+2$$ --> $$9^x + 9^{-x} = b$$. So answer to the question is YES. Sufficient.

(2) x > 0. No sufficient.

When you square both sides, I don't understand where the +2 comes from? Can you please explain?

Hello dutchmen991
this is formula that used to square expression:
$$(a+b)^2 = (a+b)(a+b) = a^2+2ab+b^2$$

In our case we have
$$3^x + 3^{-x}=3^x + \frac{1}{3^{x}}$$
When we square this expression we will have:
$$(3^x + \frac{1}{3^{x}})*(3^x + \frac{1}{3^{x}})=9^x+3x∗\frac{1}{3^x}+\frac{1}{3^x}*3^x+\frac{1}{9^x}=9^x+2*3^x*\frac{1}{3^x}+9^{-x}$$
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If x is an integer, is 9^x + 9^{-x} = b ? [#permalink]

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05 May 2015, 16:04
Harley1980 wrote:
dutchmen991 wrote:
Bunuel wrote:
SOLUTION

If x is an integer, is $$9^x + 9^{-x} = b$$ ?

(1) $$3^x + 3^{-x} = \sqrt{b + 2}$$ --> square both sides --> $$9^x+2*3^x*\frac{1}{3^x}+9^{-x}=b+2$$ --> $$9^x + 9^{-x} = b$$. So answer to the question is YES. Sufficient.

(2) x > 0. No sufficient.

When you square both sides, I don't understand where the +2 comes from? Can you please explain?

Hello dutchmen991
this is formula that used to square expression:
$$(a+b)^2 = (a+b)(a+b) = a^2+2ab+b^2$$

In our case we have
$$3^x + 3^{-x}=3^x + \frac{1}{3^{x}}$$
When we square this expression we will have:
$$(3^x + \frac{1}{3^{x}})*(3^x + \frac{1}{3^{x}})=9^x+3x∗\frac{1}{3^x}+\frac{1}{3^x}*3^x+\frac{1}{9^x}=9^x+2*3^x*\frac{1}{3^x}+9^{-x}$$

Thanks for the reply. I didn't realize this was difference of squares because I didn't know how to deal with the negative exponent. Is this the same concept being tested in the following two problems?

is-5-k-less-than-144719.html
if-2-x-2-x-2-3-2-13-what-is-the-value-of-x-130109.html
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Re: If x is an integer, is 9^x + 9^{-x} = b ? [#permalink]

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05 May 2015, 16:21
dutchmen991 wrote:

Thanks for the reply. I didn't realize this was difference of squares because I didn't know how to deal with the negative exponent. Is this the same concept being tested in the following two problems?

is-5-k-less-than-144719.html
if-2-x-2-x-2-3-2-13-what-is-the-value-of-x-130109.html

Yeah, all this tasks tests understanding of work with exponents/powers.
You can search them by tag:
search.php?search_id=tag&tag_id=60
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Re: If x is an integer, is 9^x + 9^{-x} = b ? [#permalink]

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22 May 2016, 15:24
Thought process here:

question is asking : is 9^x + 9^-x = b?

so we need to figure out how to make this happen from the data

1) 3^x + 3^-x = √b+2)
square both sides
(3^x + 3^-x)(3^x + 3^-x) = b+2
9^x + 9^-x + 2 = b + 2
subtract 2 = 9^x + 9^-x = b. SUFFICIENT

2)x > 0
This could give you multiple answers, so cannot work
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Re: If x is an integer, is 9^x + 9^{-x} = b ? [#permalink]

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22 May 2016, 15:24
Thought process here:

question is asking : is 9^x + 9^-x = b?

so we need to figure out how to make this happen from the data

1) 3^x + 3^-x = √b+2)
square both sides
(3^x + 3^-x)(3^x + 3^-x) = b+2
9^x + 9^-x + 2 = b + 2
subtract 2 = 9^x + 9^-x = b. SUFFICIENT

2)x > 0
This could give you multiple answers, so cannot work
Re: If x is an integer, is 9^x + 9^{-x} = b ?   [#permalink] 22 May 2016, 15:24
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