What is the smallest integer x such that x^14 3^34

Question

What is the smallest integer x such that

x^{14} > 3^{33} + 3^{34} + 3^{33} + 3^{34} + 3^{33}  ?

(A) 16
(B) 18
(C) 19
(D) 27
(E) 28

rajdeep212003 · Accepted Answer

sry i did a silly mistake over there 16 > 3^3 its wrong the actual analogy for the second part should be the idea is to avoid exact calculation and just compare powers.

We have:
x^14 > 3^35

So,
x > 3^(35/14)

Now instead of calculating this exactly, we approximate:

35/14 ≈ 2.5

So,
x > 3^2.5

And we know:
3^2 = 9 
3^3 = 27

So 3^2.5 lies between 9 and 27 (closer to ~15–16 range)

Now checking options, the smallest integer satisfying this is:

x = 16

Hope this clarifies the second part.

EvenPort · Answer

X^14 > 3^35
X^2 > 3^5
Therefore, X^2 is larger than 243. Thus, smallest integer is 16 that satisfies this.

rajdeep212003 · Answer

x^14 > 3^33 + 3^34 + 3^33 + 3^34 + 3^33

we don’t need to calculate everything

take 3^33 common

= 3^33(1 + 3 + 1 + 3 + 1)

= 3^33(9)

= 3^35

so,

x^14 > 3^35
no need to calculate exact value just look at the options here
here is the logic although 16 > 3^3 so 35 /3 = 11.9 so we get 16^14 > (3^3)^11.9 or 9^11.9

Final Answer: 16

— Rajdeep

kevincan · Answer

I love the first part of your explanation! Can you elaborate on the second ?

snowinmay · Answer

X^14 > 3^35
X^2 > 243
16^2 = 256
X must be 16

What is the smallest integer x such that x^14 > 3^34

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