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D01-05 05 Jul 2025, 04:24
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I like the solution - it’s helpful.
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D01-05 06 Aug 2025, 11:39
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D01-05 04 Sep 2025, 07:23
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D01-05 13 Oct 2025, 11:52
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D01-05 31 Oct 2025, 04:56
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This is a great question that’s helpful for learning and I like the solution - it’s helpful.
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D01-05 30 Nov 2025, 02:06
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I did not quite understand the solution. from 20^4 to 16*10^4 -> how did it connect??
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D01-05 30 Nov 2025, 04:41
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sunshineeee
I did not quite understand the solution. from 20^4 to 16*10^4 -> how did it connect??
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20^4 = (2*10)^4 = 2^4 * 10^4 = 16 * 10^4.
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D01-05 24 Apr 2026, 02:49
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D01-05 10 Jun 2026, 23:10
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The skill is not "comparing powers."
The skill is:
Can I rewrite both expressions so that ONE thing becomes the same?
When one thing becomes the same, the comparison becomes trivial.
Think of a checklist.
---------------------

Pattern 1: Make the exponents the same
Whenever you see:
a^m vs b^n
ask:
Can I rewrite one side so both exponents become equal?
Example:
999^12 vs 20^24
Notice:
24 = 2 × 12
So rewrite:
20^24 = (20^2)^12 = 400^12
Now compare:
999^12 vs 400^12
Same exponent ⇒ compare bases.
This is usually the FIRST thing to check.
-----------------------------------------

Pattern 2: Make the bases the same
Example:
8^10 vs 4^15
Notice:
8 = 2^3, 4 = 2^2
Rewrite:
(2^3)^10 = 2^30
(2^2)^15 = 2^30
Same base ⇒ compare exponents.
Result: equal.
--------------

Pattern 3: Compare using a common benchmark
A favorite benchmark is 10.
Example:
999^12
Since:
999 < 1000 = 10^3
999^12 < 10^36
Example:
777^10
Since:
777 < 1000
777^10 < 1000^10 = 10^30
Now both are compared to powers of 10.
--------------------------------------

Pattern 4: Take roots
Suppose:
999^12 vs 20^24
Take the 12th root of both sides.
Since roots preserve order for positive numbers:
999 vs 20^2
999 vs 400
Done.
Many students don't realize this is exactly the same logic as Pattern 1.
------------------------------------------------------------------------

Pattern 5: Compare growth by converting to the same exponent
Example:
15^20 vs 20^24
Notice:
gcd(20,24) = 4
Take fourth roots:
15^5 vs 20^6
Now comparison is much easier.
------------------------------

Pattern 6: If exponents are equal, NEVER expand
Example:
999^12 vs 777^12
Same exponent.
Immediately compare bases:
999 > 777
Done.
This should become automatic.
-----------------------------

Pattern 7: If bases are equal, NEVER calculate
Example:
3^40 vs 3^41
Same base.
Compare exponents.
41 > 40
Done.
-----

Pattern 8: Even powers remove negatives
Example:
(-20)^24
Immediately think:
20^24
because the exponent is even.
This often simplifies comparisons.
----------------------------------

Mental Flowchart
Step 1:
Are exponents already equal?
→ Compare bases.
Step 2:
Are bases already equal?
→ Compare exponents.
Step 3:
Can I make exponents equal?
→ Rewrite.
Example:
20^24 → 400^12
Step 4:
Can I make bases equal?
→ Rewrite.
Example:
8^10 → 2^30
4^15 → 2^30
Step 5:
Can I compare both to a benchmark (usually 10, 100, or 1000)?
Example:
777^10 < 1000^10 = 10^30
------------------------

For GMAT and aptitude exams, about 80–90% of exponent comparison questions are solved by one of these three moves:

1. Same exponent → compare bases.
2. Same base → compare exponents.
3. Rewrite to create the same exponent or the same base.
Train your eyes to ask:
"What can I make the same?"
That's usually the entire game.
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