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What is the value of x ?

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GMATH Teacher
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What is the value of x ?  [#permalink]

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06 Feb 2019, 08:17
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Difficulty:

65% (hard)

Question Stats:

59% (02:07) correct 41% (02:56) wrong based on 39 sessions

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GMATH practice exercise (Quant Class 14)

What is the value of $$x$$ ?

$$\left( 1 \right)\,\,\,x > 0$$
$$\left( 2 \right)\,\,\,{8^x} - 12 \cdot {4^x} = 64 \cdot {2^x}$$

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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
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Re: What is the value of x ?  [#permalink]

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06 Feb 2019, 23:56
Shouldn't answer be c? option 2 is a cubic having 3 roots : -4,0,16. Thus only with point 1 can we narrow down the answer to 4.
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What is the value of x ?  [#permalink]

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07 Feb 2019, 00:14
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Mai20 wrote:
Shouldn't answer be c? option 2 is a cubic having 3 roots : -4,0,16. Thus only with point 1 can we narrow down the answer to 4.

Mai20

It is indeed B,

have a look here

$$2^x * 2^x * 2^x - 2^2 * 3* 2^x = 64 * 2^x$$

take $$2^x$$ common from LHS and cancel it

$$2^x * 2^x - 12 * 2^x = 64$$

$$2^x$$($$2^x$$ - 12) = 64

$$2^x$$ - 12 =$$2^{6-x}$$

only x = 4 will satisfy the above

$$2^4$$ - 12 = $$2^2$$

4 = 4

B
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GMATH Teacher
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Re: What is the value of x ?  [#permalink]

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07 Feb 2019, 06:33
fskilnik wrote:
GMATH practice exercise (Quant Class 14)

What is the value of $$x$$ ?

$$\left( 1 \right)\,\,\,x > 0$$
$$\left( 2 \right)\,\,\,{8^x} - 12 \cdot {4^x} = 64 \cdot {2^x}$$

$$? = x$$

$$\left( 1 \right)\,\,\,x > 0\,\,\,\left\{ \matrix{ \,{\rm{Take}}\,\,x = 1\,\,\,\, \Rightarrow \,\,\,? = 1 \hfill \cr \,{\rm{Take}}\,\,x = 2\,\,\,\, \Rightarrow \,\,\,? = 2 \hfill \cr} \right.$$

$$\left( 2 \right)\,\,\,{\left( {{2^x}} \right)^3} - 12 \cdot {\left( {{2^x}} \right)^2} - 64 \cdot \left( {{2^x}} \right) = 0\,\,\,\,\,\mathop \Rightarrow \limits^{z\, = \,{2^x}\,\, > \,\,0} \,\,\,\,\,z\left( {{z^2} - 12z - 64} \right) = 0\,\,\,\,\,\left( * \right)$$

$$\left( * \right)\,\,\,\, \Rightarrow \,\,\,z\left( {z - 16} \right)\left( {z + 4} \right) = 0\,\,\,\,\,\mathop \Rightarrow \limits^{z\, > \,0} \,\,\,\,\,z = 16\,\,\,\,\, \Rightarrow \,\,\,\,\,{2^x} = 16\,\,\,\,\, \Rightarrow \,\,\,\,\,x\,\,{\rm{unique}}\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.$$

The correct answer is therefore (B).

We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
Re: What is the value of x ?   [#permalink] 07 Feb 2019, 06:33
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