Norway Math Olympiad Question | You should be able to solve this!
TLDRIn this educational video, the presenter tackles the mathematical problem of calculating 2 to the power of 18 minus 1. They break down the expression using the power of a power rule, transforming it into a product of (2 to the power of 9 plus 1) and (2 to the power of 9 minus 1). Recognizing 2 to the power of 9 as 512, the presenter then applies the FOIL method to the binomials (513 and 511), resulting in the final answer of 262,144. The video is an excellent resource for those looking to understand exponent rules and algebraic manipulation.
Takeaways
- 🔢 The problem presented is to solve 2^18 - 1.
- 📐 The solution involves expressing 2^18 as 2^9 × 2^9.
- 🧩 The identity a^(n × m) = (a^n)^m is used to simplify the expression.
- 🔄 The difference of squares identity a^2 - b^2 = (a + b)(a - b) is applied to 2^9 + 1 and 2^9 - 1.
- 💡 The value of 2^9 is known to be 512, which is substituted into the equation.
- 🔄 The expression simplifies to (512 + 1) × (512 - 1).
- 📈 The numbers are broken down into 513 × 511 for easier multiplication.
- 📘 The FOIL method (First, Outer, Inner, Last) is used to multiply the binomials.
- 📊 The multiplication results in 250,000 + 12,000 + 143.
- 🎯 The final answer is 2^18 - 1 = 262,626,143.
Q & A
What is the problem being solved in the video?
-The problem being solved in the video is 2 to the power of 18 minus 1.
How does the presenter approach solving the problem?
-The presenter approaches the problem by splitting the expression into 2 to the power of 9 times 2 to the power of 9 minus 1, then using the identity (a^n) * (a^m) = a^(n+m).
What identity is used in the solution process?
-The identity used in the solution process is (a^n) * (a^m) = a^(n+m).
What is the value of 2 to the power of 9?
-2 to the power of 9 is equal to 512.
How is the expression 2 to the power of 9 plus 1 times 2 to the power of 9 minus 1 simplified?
-The expression is simplified by substituting 2 to the power of 9 with 512, resulting in (512 + 1) * (512 - 1).
What is the result of 512 plus 1 and 512 minus 1?
-512 plus 1 is 513, and 512 minus 1 is 511.
What method is used to multiply the two brackets (513 and 511)?
-The presenter uses the FOIL method to multiply the two brackets.
What is the final result of the multiplication using the FOIL method?
-The final result of the multiplication using the FOIL method is 262,626.
What is the final answer to the problem 2 to the power of 18 minus 1?
-The final answer to the problem 2 to the power of 18 minus 1 is 262,626.
How does the presenter ensure clarity in the solution process?
-The presenter ensures clarity by breaking down the problem step by step, using identities, and explaining each step clearly.
What is the significance of using the identity (a^n) * (a^m) = a^(n+m) in this problem?
-Using the identity (a^n) * (a^m) = a^(n+m) simplifies the problem by allowing the exponents to be added, which is easier to handle than multiplying large numbers.
Outlines
📐 Solving 2 to the Power of 18 Minus 1
In this segment, the speaker introduces the problem of solving 2^18 - 1. They begin by explaining how to split the exponent into 2^9 * 2, as 18 equals 9 * 2. This step is based on the power of powers rule, where (a^n)^m equals a^(n*m). The speaker prepares to apply this identity and solve for the expression.
🧩 Applying Algebraic Identities to Simplify
The speaker notes that the expression 2^18 - 1 is similar to the algebraic identity (a^2 - b^2) = (a + b)(a - b). Using this identity, they express the equation as (2^9 + 1)(2^9 - 1). They then calculate 2^9 as 512, leading to the updated expression (513)(511).
🧮 Breaking Down the Multiplication
Next, the speaker decides to split 513 and 511 into smaller, more manageable parts: 500 + 13 and 500 + 11, respectively. This makes it easier to apply the FOIL method (First, Outer, Inner, Last) for multiplying the two terms. They outline each multiplication step, starting with 500 * 500.
🔢 Performing the Calculations
The multiplication begins with 500 * 500, yielding 250,000. Then, the speaker adds 500 multiplied by 13 and 11, resulting in 500 * 24, and finally, calculates 13 * 11 to get 143. These values are summed to simplify the expression further.
📝 Final Sum and Conclusion
The final step is summing 250,000, 12,000 (from 500 * 24), and 143, which results in 262,143. The speaker concludes that 2^18 - 1 equals 262,143, and ends the video with a brief farewell and hopes that the explanation was helpful.
Mindmap
Keywords
💡Exponentiation
💡2^18 - 1
💡Power rule
💡Difference of squares
💡Substitution
💡FOIL method
💡Multiplication of binomials
💡Mental math
💡Factoring
💡Math Olympiad
Highlights
Solving 2 to the power of 18 minus 1
Splitting the expression using exponent rules
Using the identity a^(n*m) = (a^n)^m
Rewriting the expression as (2^9)^2 - 1
Applying the difference of squares identity
Calculating 2^9 and its value as 512
Substituting 512 into the expression
Simplifying to (512+1)(512-1)
Using the FOIL method to expand the brackets
Breaking down numbers into simpler components
Calculating 500*500 and other terms
Adding up the results to get the final answer
Final answer of 262,144
Thanking viewers and closing the session