If 2^(8x) = 640000, then what is the value of 2^(2x−2)?

Question

If  2^{(8x)} = 640000 , then what is the value of  2^{(2x−2)} ?

A. 5√2
B. 10
C. 10√2
D. 80√2
E. 160

Bunuel · Accepted Answer

Step 1:

2^{(8x)} = 640000 ;

(2^{2x})^4 = 2^{10}*5^4 ;

2^{2x} = 2^{(\frac{5}{2})}*5 ;

2^{2x} = \sqrt{2^5}*5 ;

2^{2x} = 20\sqrt{2} .

Step 2:

2^{(2x−2)}=\frac{2^{(2x)}}{2^2} .

Step 2:

Substitute  2^{2x} = 20\sqrt{2}  into  \frac{2^{(2x)}}{2^2} .:

\frac{20\sqrt{2}}{2^2}=5\sqrt{2} .

Answer: A.

Hope it's clear.

0akshay0 · Answer

2^{(8x)} = 640000 = 2^6*10^4 
 2^{(2x)} = 2^{6/4}*10^{4/4} 
 2^{(2x)} = 2^{3/2}*10 
 2^{(2x)} = 2\sqrt{2}*10 = 20\sqrt{2}

2^{(2x-2)} = 20\sqrt{2}/4 = 5\sqrt{2}

Hence Option A is correct

Spartan85 · Answer

In the step 2^(2x)=20√2 is OK Now 2^(2x−2)=2^(2X) / 2^2 = 20√2 / 4 = 5√2 :?:

Bunuel · Answer

Similar questions to practice: https://gmatclub.com/forum/if-3-6x-8-10 ... 98777.html https://gmatclub.com/forum/if-3-6x-810- ... 54347.html 8. Exponents and Roots of Numbers Theory Exponents and Roots on the GMAT: Tips and hints Cyclicity and the remainders on the GMAT Questions Units digits, exponents, remainders problems Tough and tricky DS exponents and roots questions with detailed solutions Tough and tricky PS exponents and roots questions with detailed solutions Exponents DS Questions Exponents PS Questions Roots DS Questions Roots PS Questions Check below for more: ALL YOU NEED FOR QUANT ! ! ! Ultimate GMAT Quantitative Megathread Hope it helps.

BrentGMATPrepNow · Answer

Another approach:

Given: 2^(8x) = 640000
Raise both sides to the power 1/2 (aka take square root of both sides) to get:  ^(1/2) = 640000^(1/2)
Simplify to get: 2^(4x) = 800
Raise both sides to the power 1/2 (again) to get:  ^(1/2) = 800^(1/2)
Simplify: 2^2x = √800
Simplify: 2^2x = 20√2
Divide both sides by 2² (aka 4) to get: (2^2x)/2² = (20√2)/4
Simplify both sides to get: 2^(2x - 2) = 5√2

Answer: A

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chetan2u · Answer

Another approach, although most of these approaches is solving the equation..

Let  2^{(2x−2)}=y , so  y^4=2^{(2x−2)*4}=2^{8x-8}=\frac{2^{8x}}{2^8}=\frac{640000}{2^6*2^2}=\frac{10000}{2^2} 
so  y^4=\frac{10000}{2^2}=\frac{10^4}{2^2}....y=\frac{10}{\sqrt{2}}=5\sqrt{2}

A

ScottTargetTestPrep · Answer

We need to determine the value of (2^(2x))/2^2 = (2^(2x))/4

If we take the square root of the given equation, we are left with:

2^(4x) = 800

If we take the square root of 2^(4x) = 800, we have:

2^(2x) = √800

2^(2x) = √800 = √400 x √2 = 20√2

Thus, (2^(2x))/4 = 20√2/4 = 5√2

Answer: A

Pritishd · Answer

While the solutions posted above are all elegant, I am sharing here the way to solve backwards from answer choices. This will be helpful in case one is not able to think of the direct approach.

1.  2^{(2x−2)}  can be re-written as  2^{2x}/2^2  (----> 1)
2. Similarly,  2^{(8x)} = 640,000  can be re-written as  2^{2x(4)}   =   640,000  to make it inline with the numerator i.e.  2^{2x}  from point  1  (----> 2)

Let's start with option A:
 2^{(2x−2)}  =  2^{2x}/2^2  =  5\sqrt{2} 
 2^{2x}   =   5\sqrt{2} * 2^2  =  20\sqrt{2}

Plugging the value of  2^{2x}  into  2^{2x(4)}  from (----> 2)
 (20\sqrt{2})^4   =   640,000

As we get the answer from the first option itself, we do not need to test the other answer options

Ans.  A

Kritisood · Answer

in such questions keep an eye on what is being asked.
here:  2^{(2x−2)}

given:  2^{(8x)} = 640000  =>  2^{8x}=2^{10}*5^{4}

First, we need 2x in the power, so:
 2^{2x}=2^{5/2}*5 
now we need 2x-2
 2^{2x-2}=2^{1/2}*5

ans: A. 5√2

GMATGuruNY · Answer

\frac{2^{(8x)}}{2^8} = \frac{640000}{2^8} 
 2^{(8x-8)} = \frac{2^6*2^4*5^4}{2^8} 
 2^{(8x-8)} = 2^2*5^4 
 2^{(8x-8)}^{\frac{1}{4}} = (2^2*5^4)^{\frac{1}{4}} 
 2^{(2x-2)} = \sqrt{2}*5

A

tejasvkalra · Answer

2^8x = 640000 = 2^6 * 10^4

2^2x = 2^(8x/4) = 2^(6/4) * 10 = 2^(3/2) * 10

now , 2^(2x-2) = 2^2x / 2 = 2^ (3/2) * 10 / 2 = 2^(1.5-2) / 10 = 10/√2 = 5√2

Kinshook · Answer

If  2^{(8x)} = 640000 , then what is the value of  2^{(2x−2)} ?

(2^{2x})^4 = 640000 
 2^{2x} = (640000)^{1/4} = 20\sqrt{2} 
 2^{(2x−2)} = 2^{2x}*2^{-2} = 5\sqrt{2}

IMO A

If 2^(8x) = 640000, then what is the value of 2^(2x−2)?

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If 2^(8x) = 640000, then what is the value of 2^(2x−2)?

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