Comparing Powers by Simplifying Exponents
Comparing powers by simplifying exponents is a Grade 4 AMC Math skill where students determine which of two large power expressions is greater by rewriting them with a common base or exponent. This technique avoids computing enormous numbers directly, a common strategy in AMC competition math.
Key Concepts
Property To compare two exponential numbers like $a^m$ and $b^n$, we can simplify their exponents by taking a common root. Let $k = \text{gcf}(m, n)$. The comparison between $a^m$ and $b^n$ is equivalent to comparing their $k$ th roots. $$a^m b^n \iff (a^m)^{\frac{1}{k}} (b^n)^{\frac{1}{k}} \iff a^{\frac{m}{k}} b^{\frac{n}{k}}$$ This reduces the exponents to smaller, more manageable numbers.
Examples Example 1 Which is greater: 3^400 or 5^240? The exponents are 400 and 240. The greatest common factor is gcf(400, 240) = 80. We compare (3^400)^(1/80) and (5^240)^(1/80). This simplifies to comparing 3^5 and 5^3, which is 243 vs 125. Since 243 125, we have 3^400 5^240.
Example 2 Which is greater: $5^{48}$ or $10^{36}$? The exponents are $48$ and $36$. The greatest common factor is $\text{gcf}(48, 36) = 12$. We compare $5^{\frac{48}{12}}$ and $10^{\frac{36}{12}}$. This simplifies to comparing $5^4$ and $10^3$, which is $625$ vs $1000$. Since $625 < 1000$, we have $5^{48} < 10^{36}$.
Common Questions
How do you compare powers like 2^30 and 3^20?
Rewrite with the same exponent: 2^30 = (2^3)^10 = 8^10 and 3^20 = (3^2)^10 = 9^10. Since 9 > 8, we get 3^20 > 2^30.
Why is it helpful to simplify exponents before comparing?
Simplifying to a common exponent transforms the comparison into comparing the bases, which is much easier than computing the actual values.
Can you compare powers with different bases and exponents?
Yes. Rewrite each power by adjusting base or exponent to a common form. The key is making the comparison equivalent but simpler.
What are useful strategies for AMC exponent comparison problems?
Rewrite to a common exponent, use logarithms (log a^m vs log b^n = m log a vs n log b), or find a convenient common base.