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# If x and y are real numbers and a^2x*a^2y = 81, what is the value of x

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Joined: 02 Sep 2009
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If x and y are real numbers and a^2x*a^2y = 81, what is the value of x  [#permalink]

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11 Feb 2019, 03:02
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62% (01:28) correct 38% (01:26) wrong based on 136 sessions

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If x and y are real numbers and $$a^{2x}*a^{2y} = 81$$, what is the value of x + y?

(1) a = 3
(2) x = y

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If x and y are real numbers and a^2x*a^2y = 81, what is the value of x  [#permalink]

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11 Feb 2019, 03:09
1
Bunuel wrote:
If x and y are real numbers and $$a^{2x}*a^{2y} = 81$$, what is the value of x + y?

(1) a = 3
(2) x = y

We'll first simplify our data so we know what we have to look for.
This is a Precise approach.

$$a^{2x}*a^{2y} = a^{2x+2y} = a^{2(x+y)}=(a^{x+y})^2$$
Therefore $$a^{x+y}$$ equals 9 or -9.

If a= 3 then x + y must equal 2, so (1) is Sufficient.
Knowing that x = y gives no information about their sum, so (2) is Insufficient.

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Re: If x and y are real numbers and a^2x*a^2y = 81, what is the value of x  [#permalink]

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11 Feb 2019, 03:12
Bunuel wrote:
If x and y are real numbers and $$a^{2x}*a^{2y} = 81$$, what is the value of x + y?

(1) a = 3
(2) x = y

$$a^{2x}*a^{2y} = 81$$

$$2^{2x + 2y} = 81$$

$$81 =9^2$$

$$81 = 3^4.$$

Statement 1: a = 3. Sufficient.

$$3^{2x + 2y} = 3^4$$

2x + 2y = 4

x + y = 2.

Statement 2:

x=y.

$$a^{2x + 2x} = 81$$

$$a^{4x} = 3^4$$

or

$$a^{4x} = 9^2$$

2 Options.

NOT sufficient.

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If x and y are real numbers and a^2x*a^2y = 81, what is the value of x  [#permalink]

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31 Mar 2020, 09:43
KSBGC wrote:
Bunuel wrote:
If x and y are real numbers and $$a^{2x}*a^{2y} = 81$$, what is the value of x + y?

(1) a = 3
(2) x = y

$$a^{2x}*a^{2y} = 81$$

$$2^{2x + 2y} = 81$$

$$81 =9^2$$

$$81 = 3^4.$$

Statement 1: a = 3. Sufficient.

$$3^{2x + 2y} = 3^4$$

2x + 2y = 4

x + y = 2.

Statement 2:

x=y.

$$a^{2x + 2x} = 81$$

$$a^{4x} = 3^4$$

or

$$a^{4x} = 9^2$$

2 Options.

NOT sufficient.

Regarding statment 2 , in your second option X need to be in that case 1/2 , and the question did state "Real number " can you please explain that one ?
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Re: If x and y are real numbers and a^2x*a^2y = 81, what is the value of x  [#permalink]

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31 Mar 2020, 14:05
Bunuel wrote:
If x and y are real numbers and $$a^{2x}*a^{2y} = 81$$, what is the value of x + y?

(1) a = 3
(2) x = y

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.

$$a^{2x} \cdot a^{2y} = 81$$
$$⇔ a^{2x+2y} = 3^4$$
$$⇔ a^{2(x+y)} = 3^4$$

Then, if $$a = 3$$, then we have $$2(x+y) = 4$$ or $$x+y = 2$$.
Thus condition 1) alone is sufficient.

Condition 2)
If $$x = 1, y = 1$$, then we have $$a^4 = 81$$, which implies $$a = 3$$ or $$-3$$.

Since condition 2) does not yield a unique solution, it is not sufficient.

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Re: If x and y are real numbers and a^2x*a^2y = 81, what is the value of x   [#permalink] 31 Mar 2020, 14:05