Norway Math Olympiad Question | You should be able to solve this!
Summary
TLDRIn 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.
Outlines
🔢 Solving 2^18 - 1 Using Algebraic Identities
The video script begins with the presenter introducing a mathematical problem: calculating 2 to the power of 18 minus 1. The presenter splits the expression using the identity for a power of a product, rewriting it as (2^9) * (2^9) - 1. Recognizing this as a form of the identity (a+b)(a-b) = a^2 - b^2, the presenter simplifies the problem to (2^9 + 1) * (2^9 - 1). Knowing that 2^9 equals 512, the presenter substitutes this value into the equation, resulting in 513 * 511. The presenter then breaks down the multiplication using the FOIL method (First, Outer, Inner, Last), multiplying the terms and adding the results to get the final answer of 262,143. The presenter concludes by thanking the viewers for watching and hopes the explanation is helpful.
Mindmap
Keywords
💡Power of a number
💡Exponentiation rule
💡Identity
💡Substitution
💡FOIL method
💡Binomial
💡Multiplication
💡Addition
💡Calculation
💡Result
💡Algebraic manipulation
Highlights
Introduction to solving the equation 2 to the power of 18 minus 1.
Splitting the equation using the identity a to the power of n times a to the power of m equals a to the power of n+m.
Rewriting the equation as 2 to the power of 9 times 2 to the power of 9 minus 1.
Using the identity a squared plus b squared equals a plus b times a minus b.
Substituting 2 to the power of 9 with 512.
Calculating 512 plus 1 and 512 minus 1.
Simplifying the equation to 513 times 511.
Breaking down the numbers into 500 plus 13 and 500 plus 11.
Applying the FOIL method (First, Outside, Inside, Last) to multiply the brackets.
Calculating 500 times 500.
Calculating 500 times 13 plus 11.
Calculating 500 times 13 plus 500 times 11 plus 13 times 11.
Summing up the results to get 250,000 plus 12,000 plus 143.
Final calculation resulting in 262,143.
Conclusion that 2 to the power of 18 minus 1 equals 262,143.
Thanking the audience and ending the explanation.
Transcripts
hi everyone in this we're gonna solve 2
to the power of 18 minus 1 equals what
so first of all we're going to split
this we are going to write it as 2 to
the power of 9 times 2 is 18 so we know
that 9
times 2 so we can write it like this 2
to the power of 9 times 2 is 18 minus 1
so what I have used here a to the power
of n whole to the power of M is equal to
a to the power of n times m
so here you have the product of the
power so you can split it in this way
now
what does it look like this is in a form
of an identity a square point is b
square which is equal to
a plus b times a minus B now we're going
to solve for that
so we have 2 to the power of 9 plus 1
times 2 to the power of 9 minus 1. and
then as we know 2 to the power of 9 is
equal to 512 so we're going to
substitute that 512 plus 1
times 512 minus 1. so 512 plus 1 is 513
and 512 minus 1 is 511.
so now we are just gonna
split this
both
into
500 plus 13 times 500 plus 11.
and now
we're gonna use the foil method and
multiply the two brackets so we have 500
times 500 so we're going to write that
500
times 500
then we have 500 times 11 500 times 13
so we will write it like this plus 500
multiplied by 13 plus 11.
and then
the last term plus 13 times 11. now 500
times 500 is
250
000.
Plus
500 times 13 plus 11 is 24. plus 13
times 11 is 143.
now
250
000
plus 500 times 24 is just 12
000.
plus 143 now let's just add them up
together so we have two hundred and
fifty thousand
plus twelve thousand plus
143 so when you add them up you get
three four one
two
six two so you have the answer of
2 to the power of 18 minus 1 is equal to
2 6 262 143.
thank you so much for watching hope this
is helpful bye bye take care
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