Depth · one concept from first principles · step 2 of 4
Match the exponents, then compare bases
Of the four numbers below, find the largest one and the smallest one, in that order.
, , ,
Expressions
Your answer
How to solve
Strategy Change Focus / Count the Complement — Instead of comparing bases with different exponents, change focus to the exponents: 40, 30, 20, 10 are all multiples of 10. Rewrite every number as something to the 10th power, and then only the bases need comparing.
1STEP 1
Pick a common exponent
All four exponents are multiples of 10, so aim for the form ( )¹⁰.
40=4×10, 30=3×10, 20=2×10
A shared exponent turns four comparisons into comparing four bases.
8.EE.A.1Change Focus Count The Complement2STEP 2
Rewrite each power
Use the power-of-a-power rule in reverse.
2⁴⁰=(2⁴)¹⁰=16¹⁰, 3³⁰=(3³)¹⁰=27¹⁰, 5²⁰=(5²)¹⁰=25¹⁰
(a^m)ⁿ=a^(mn), read backwards.
8.EE.A.1Organize Information In More Ways3STEP 3
Compare the bases
With the same positive exponent, a bigger positive base gives a bigger power.
15 ＜ 16 ＜ 25 ＜ 27 → 15¹⁰ ＜ 16¹⁰ ＜ 25¹⁰ ＜ 27¹⁰
Multiplying ten copies of a bigger number gives a bigger product.
6.EE.A.1Organize Information In More Waysgives .
Why?
With the same exponent , the bigger positive base always gives the bigger power, so the order of the bases carries straight over to their tenth powers.
🧱A bigger base makes a bigger power (natural exponent)For the same natural-number exponent n, each of the n factors of a bigger positive base is bigger, so if a > b > 0 then a^n > b^n; therefore a^n = b^n with a, b positive forces a = b.
4STEP 4
Translate back
Replace each base-to-the-tenth with the original number.
15¹⁰ ＜ 2⁴⁰ ＜ 5²⁰ ＜ 3³⁰
16¹⁰ is 2⁴⁰, and so on.
8.EE.A.1Organize Information In More WaysAnswer
3³⁰, 15¹⁰
Rough sizes agree: 2⁴⁰≈1.1×10¹², 3³⁰≈2.1×10¹⁴, 5²⁰≈9.5×10¹³, 15¹⁰≈5.8×10¹¹. The biggest is 3³⁰ and the smallest is 15¹⁰.
Takeaway
Different bases and exponents? Make the exponents match -- then the bases tell you everything.
- Pick a common exponent
- Rewrite each power
- Compare the bases
- Translate back
Where next?
Another one like thissuggested
Step back and solidifysuggested
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