Comparing Powers by Simplifying Exponents
Comparing Powers by Simplifying Exponents is a skill on Pengi from Lesson 2: Which is Greater? in AoPS: Introduction to Algebra (AMC 8 & 10).
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
What is Comparing Powers by Simplifying Exponents?
Comparing Powers by Simplifying Exponents is a skill on Pengi from Lesson 2: Which is Greater? in AoPS: Introduction to Algebra (AMC 8 & 10).
What grade level is Comparing Powers by Simplifying Exponents for?
Comparing Powers by Simplifying Exponents is part of the Grade 4 Amc_math curriculum, covered in AoPS: Introduction to Algebra (AMC 8 & 10). It is designed for students studying Amc_math at the Grade 4 level.
How can I learn Comparing Powers by Simplifying Exponents?
Pengi offers an AI-guided lesson for Comparing Powers by Simplifying Exponents that walks you through the key concepts step by step. The lesson is aligned to AoPS: Introduction to Algebra (AMC 8 & 10) so the content matches what you see in class.
How do I practice Comparing Powers by Simplifying Exponents?
After learning the concept, use the Practice mode to work through targeted exercises on Comparing Powers by Simplifying Exponents. The AI adapts to your level and gives feedback on each answer so you can identify and fix mistakes.
Which textbook covers Comparing Powers by Simplifying Exponents?
Comparing Powers by Simplifying Exponents is covered in AoPS: Introduction to Algebra (AMC 8 & 10), specifically in Chapter 9: Introduction to Inequalities under Lesson 2: Which is Greater?. Pengi's lesson is aligned directly to this textbook so you can follow along with your class.
Is Comparing Powers by Simplifying Exponents free to study on Pengi?
Yes, the core Learn and Practice modes for Comparing Powers by Simplifying Exponents are available for free on Pengi. No credit card is required to start studying.