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10 Mar 2026, 14:14
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B
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Difficulty:
75%
(hard)
Question Stats:
51% (02:39) correct
49%
(02:22)
wrong
based on 41
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Let
a = 2^80 * 3^16
b = 2^16 * 7^32
c = 3^32 * 11^16
Which of the following lists a, b, and c in increasing order?
A. b < a < c
B. a < b < c
C. a < c < b
D. c < b < a
E. b < c < a
a = 2^80 * 3^16
b = 2^16 * 7^32
c = 3^32 * 11^16
Which of the following lists a, b, and c in increasing order?
A. b < a < c
B. a < b < c
C. a < c < b
D. c < b < a
E. b < c < a
10 Mar 2026, 14:43
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Key concept being tested: Comparing Exponential Expressions with Different Bases (Problem Solving) — specifically, rewriting expressions as the same exponent to compare bases directly.
The trick here is to notice that all three exponents share a common factor of 16, which lets you take the 16th root of each expression and compare the resulting base values instead.
1. Rewrite a as a perfect 16th power:
a = 2^80 * 3^16 = (2^5)^16 * (3)^16 = (32 * 3)^16 = 96^16
2. Rewrite b as a perfect 16th power:
b = 2^16 * 7^32 = (2)^16 * (7^2)^16 = (2 * 49)^16 = 98^16
3. Rewrite c as a perfect 16th power:
c = 3^32 * 11^16 = (3^2)^16 * (11)^16 = (9 * 11)^16 = 99^16
4. Now compare the bases:
96^16 vs 98^16 vs 99^16 → since 96 < 98 < 99 and all exponents are equal and positive, we get:
a < b < c
Answer: B
Common trap: Students try to compare these expressions directly by looking at the individual prime bases (2, 3, 7, 11) and get confused because they seem unrelated. The moment you spot that all exponents are multiples of 16, the path is clear — factor out the 16th power and compare the composite bases.
Takeaway: whenever you see exponential expressions with different bases, look for a common exponent you can factor out so you can compare on a level playing field.
The trick here is to notice that all three exponents share a common factor of 16, which lets you take the 16th root of each expression and compare the resulting base values instead.
1. Rewrite a as a perfect 16th power:
a = 2^80 * 3^16 = (2^5)^16 * (3)^16 = (32 * 3)^16 = 96^16
2. Rewrite b as a perfect 16th power:
b = 2^16 * 7^32 = (2)^16 * (7^2)^16 = (2 * 49)^16 = 98^16
3. Rewrite c as a perfect 16th power:
c = 3^32 * 11^16 = (3^2)^16 * (11)^16 = (9 * 11)^16 = 99^16
4. Now compare the bases:
96^16 vs 98^16 vs 99^16 → since 96 < 98 < 99 and all exponents are equal and positive, we get:
a < b < c
Answer: B
Common trap: Students try to compare these expressions directly by looking at the individual prime bases (2, 3, 7, 11) and get confused because they seem unrelated. The moment you spot that all exponents are multiples of 16, the path is clear — factor out the 16th power and compare the composite bases.
Takeaway: whenever you see exponential expressions with different bases, look for a common exponent you can factor out so you can compare on a level playing field.
