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10 Mar 2026, 09:21
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
35%
(medium)
Question Stats:
68% (01:25) correct
32%
(01:51)
wrong
based on 77
sessions
History
Date
Time
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Not Attempted Yet
If \(\sqrt{2\sqrt{2\sqrt{2}}} = 2^p\), what is the value of p?
A. 3/8
B. 3/4
C. 7/8
D. 5/4
E. 7/4
A. 3/8
B. 3/4
C. 7/8
D. 5/4
E. 7/4
10 Mar 2026, 09:33
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kevincan
Given : sqrt(2 sqrt(2 sqrt(2))) = 2^p
squaring we get,
2 sqrt(2 sqrt(2))= 2^2p
Again squaring we get
4* (2 sqrt(2)) = 2^ 4p
8 * sqrt (2) = 2^4p
Squaring again, we get
64 * 2 = 2^8p
2^6 = 64 , then LHS becomes 2^7 , and RHS = 2^8p
Equating the powers , we get 7 = 8p
Thus, p = (7/8)
Option C
10 Mar 2026, 11:15
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Key concept being tested: Fractional Exponents with Nested Radicals (Problem Solving)
The cleanest approach here is to convert everything to powers of 2 from the inside out, using the rule sqrt(x) = x^(1/2).
1. Start with the innermost radical: sqrt(2) = 2^(1/2)
2. Move to the middle layer: 2 * sqrt(2) = 2^1 * 2^(1/2) = 2^(3/2)
Now take the square root of that: sqrt(2^(3/2)) = 2^(3/4)
3. Move to the outer layer: 2 * 2^(3/4) = 2^1 * 2^(3/4) = 2^(7/4)
Now take the square root of that: sqrt(2^(7/4)) = 2^(7/8)
4. So we have 2^(7/8) = 2^p, which means p = 7/8.
Answer: C
Common trap: Many students try squaring both sides repeatedly — it works, but tracking three layers of squaring is messy and error-prone under timed conditions. The fractional exponent approach is faster and less likely to cause arithmetic slips.
The pattern worth remembering: each layer of "multiply by 2 then take sqrt" multiplies the existing exponent by 1/2 and adds 1/2. So this type of problem unravels cleanly if you stay in exponent form throughout.
Takeaway: On Problem Solving questions involving nested radicals of the same base, immediately rewrite every sqrt as a ^(1/2) and combine exponents — it turns a visually complex expression into a straightforward exponent addition problem.
The cleanest approach here is to convert everything to powers of 2 from the inside out, using the rule sqrt(x) = x^(1/2).
1. Start with the innermost radical: sqrt(2) = 2^(1/2)
2. Move to the middle layer: 2 * sqrt(2) = 2^1 * 2^(1/2) = 2^(3/2)
Now take the square root of that: sqrt(2^(3/2)) = 2^(3/4)
3. Move to the outer layer: 2 * 2^(3/4) = 2^1 * 2^(3/4) = 2^(7/4)
Now take the square root of that: sqrt(2^(7/4)) = 2^(7/8)
4. So we have 2^(7/8) = 2^p, which means p = 7/8.
Answer: C
Common trap: Many students try squaring both sides repeatedly — it works, but tracking three layers of squaring is messy and error-prone under timed conditions. The fractional exponent approach is faster and less likely to cause arithmetic slips.
The pattern worth remembering: each layer of "multiply by 2 then take sqrt" multiplies the existing exponent by 1/2 and adds 1/2. So this type of problem unravels cleanly if you stay in exponent form throughout.
Takeaway: On Problem Solving questions involving nested radicals of the same base, immediately rewrite every sqrt as a ^(1/2) and combine exponents — it turns a visually complex expression into a straightforward exponent addition problem.
