Norway Math Olympiad Question | You should be able to solve this!

Summary

In this educational video, the presenter demonstrates a method to solve the expression 2 to the power of 18 minus 1. They start by splitting the expression using the power of a product rule, then apply the identity (a+b)(a-b) = a^2 - b^2. After substituting 2 to the power of 9 with 512, they simplify the expression to 513 * 511 using the FOIL method. The final calculation results in 2,626,143, showcasing a clear step-by-step process that is both informative and engaging.

Takeaways

• 🔢 The problem presented is to solve \(2^{18} - 1\).
• 📝 The solution involves splitting the expression into \(2^{9} \times 2^9 - 1\).
• 🧩 The script uses the identity \(a^n \times a^m = a^{n+m}\) to simplify the expression.
• 🔑 It then applies the formula \((a + b)(a - b) = a^2 - b^2\) to further simplify the problem.
• 📈 The base \(2^9\) is calculated to be 512, which is substituted into the formula.
• 📝 The expression is then broken down into \((512 + 1) \times (512 - 1)\).
• 🔍 The numbers 513 and 511 are derived from adding and subtracting 1 from 512, respectively.
• 📚 The script uses the FOIL method (First, Outer, Inner, Last) to expand the expression.
• 📈 The multiplication is carried out with the numbers broken down into 500 + 13 and 500 + 11.
• 📊 The final calculation involves multiplying and adding the terms to get the result.
• 🎉 The final answer given is \(2^{18} - 1 = 262,143\).

Q & A

What is the mathematical expression being solved in the video?

The mathematical expression being solved is \(2^{18} - 1\).

How does the video split the expression \(2^{18} - 1\)?

The video splits the expression as \(2^{9} imes 2^9 - 1\), recognizing that \(2^9 imes 2\) equals \(2^{18}\).

What mathematical identity is used to simplify the expression?

The identity \(a^n imes a^m = a^{n+m}\) is used to simplify the expression.

What is the form of the identity used in the video?

The identity used is in the form \(a^2 - b^2 = (a+b)(a-b)\).

What is the value of \(2^9\) according to the video?

The value of \(2^9\) is given as 512.

How is the expression \(512 + 1\) simplified in the video?

The expression \(512 + 1\) is simplified to 513.

What is the expression for \(512 - 1\) in the video?

The expression \(512 - 1\) is simplified to 511.

What method is used to multiply the terms in the video?

The FOIL (First, Outer, Inner, Last) method is used to multiply the terms.

How is the multiplication of the terms broken down in the video?

The multiplication is broken down into \(500 imes 500\), \(500 imes 11\), \(500 imes 13\), and \(13 imes 11\).

What is the final answer given for \(2^{18} - 1\) in the video?

The final answer given is 2,626,214.