If 3^k = 16, and 2^j = 27, then kj =

Question

If  3^k  = 16, and  2^j  = 27, then kj =

A) 8
B) 9
C) 10
D) 12
D) 15

* Kudos for all correct solutions

BrentGMATPrepNow · Accepted Answer

Another approach is to  isolate the 3 in both equations . Here’s what I mean:

Given: 3^k = 16
Rewrite 16 as 2^4 to get: 3^k = 2^4
Raise both sides to the power of 1/k to get: (3^k)^(1/k) = (2^4)^(1/k)
Use power of power law to simplify:  3 = 2^(4/k)

Given: 2^j = 27
Rewrite 27 as 3^3 to get: 2^j = 3^3
Raise both sides to the power of 1/3 to get: (2^j)^(1/3) = (3^3) ^(1/3) 
Use power of power law to simplify:  2^(j/3) = 3

We now have two equations:
 3 = 2^(4/k) 
 2^(j/3) = 3

Since both equations are set equal to 3, we can write:  2^(4/k)  =  2^(j/3) 
Since the bases both equal 2, we can conclude that 4/k = j/3
Cross multiply to get: jk = (4)(3) 
So, jk = 12

Answer:  D

Cheers,
Brent

rohit8865 · Answer

GMATPrepNow

question must be 3^k=27 && 2^j=16

Any way
3^k=27=3^3 thus k=3
2^j=16=2^4 thus j=4
so kj= 3*4 =12

Ans D

BrentGMATPrepNow · Answer

The given information is correct as worded:  3^k  = 16, and  2^j  = 27
The solution to each individual equation is not an integer.

Cheers,
Brent

rohit8865 · Answer

then we can multiply both equation to get
3^k*2^j= 16*27
getting k=3 j=4
thus KJ=12

Abhishek009 · Answer

The question looks a bit odd , but I am with the above solution, answer must be (D)...

BrentGMATPrepNow · Answer

Another approach involves  approximation .

Given: 2^ j  = 27
Notice that 2^ 4  = 16 and 2^ 5  = 32
Since 27 is closer to 32 than it is to 16, we can conclude that  j  is closer to  5  than it is to  4 . 
Let's say  j ≈ 4.7

Given: 3^ k  = 16
Notice that 3^ 2  = 9 and 3^ 3  = 27
Since 16 is approximately halfway between 9 and 27, we can conclude that  k  is approximately halfway between  2  and  3 . 
Let's say  k ≈ 2.5

So,  j  k  ≈ ( 4.7 )( 2.5 )
≈ 11.75

When we check the answer choices, we see that answer choice D is the closest to 11.75

Cheers,
Brent

ankujgupta · Answer

Another approach. 3^k = 16. Take log to the base 3. k= log 3 16 = 4 log 3 2 As log a (b^k) = k log a b
2^j = 27. Take log to the base 2. j = log 2 27 = 3log 3 3
k*j = 12 * log 3 2 * log  2 3
Thus kj = 12. as log  a b * log b a =1

Mkrishnabdrr · Answer

3^k=16 and 2^J=27, we need to find kJ
3^k x 2^J=16 x 27
3^k x 2^J=2^4 x 3^3
i.e. 3^k x 2^J=3^3 x 2^4
Thus, kJ=12

Ans : D

shreyashree · Answer

The answer would be D i.e. 12.

Multiply the two and get a he respective powers of 2 and 3.

Thus k*j=12

Sent from my ONEPLUS A3003 using  GMAT Club Forum mobile app

BrushMyQuant · Answer

Given that  3^k  = 16, and  2^j  = 27 and we need to find the value of kj

3^k  = 16 =  2^4

Raising both the sides to the power of  \frac{1}{k}  we get
 3^{k/k}  =  2^{\frac{4}{k}} 
=> 3 =  2^{\frac{4}{k}}  ... (1)

2^j  = 27 =  3^3  
Raising both the sides to the power of  \frac{1}{3}  we get
 2^{\frac{j}{3}}  =  3^{\frac{3}{3}} 
=>  2^{\frac{j}{3}}  = 3 ... (2)

From (1) and (2) we get 
3 =  2^{\frac{4}{k}}  =  2^{\frac{j}{3}} 
=>  \frac{4}{k}  =  \frac{j}{3} 
=> 4 * 3 = k*j
=> kj = 12

So,  Answer will be D 
Hope it helps!

Watch the following video to learn the Basics of Exponents

https://www.youtube.com/watch?v=9la4tJI7ZiI

bumpbot · Answer

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