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.

Outlines

00:00

🔢 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.

Keywords

💡Power of a number

The 'power of a number' refers to the result of multiplying the number by itself a certain number of times, denoted as a^n where 'a' is the base and 'n' is the exponent. In the video, the power concept is central to solving the problem 2^18 - 1, where the script breaks down the expression using the power of 2.

💡Exponentiation rule

The 'exponentiation rule' (a^n * a^m = a^(n+m)) is a mathematical principle that states when multiplying two powers with the same base, you can add their exponents. The script applies this rule to simplify 2^18 - 1 into 2^9 * 2^9 - 1.

💡Identity

In mathematics, an 'identity' is a property or equation that holds true for all values within a specified set. The script uses the identity (a^2 - b^2 = (a+b)(a-b)) to transform the expression 2^18 - 1 into a more manageable form.

💡Substitution

The process of 'substitution' involves replacing a part of an equation with an equivalent expression. In the script, the value of 2^9 is substituted with 512 to simplify the calculation of 2^18 - 1.

💡FOIL method

The 'FOIL' method (First, Outer, Inner, Last) is a technique for multiplying two binomials. The script uses this method to expand (512 + 1)(512 - 1) into a more straightforward calculation.

💡Binomial

A 'binomial' is an algebraic expression with two terms. The script refers to expressions like (512 + 1) and (512 - 1) as binomials when applying the FOIL method.

💡Multiplication

The term 'multiplication' is a fundamental arithmetic operation where two numbers are combined to give a product. The script uses multiplication extensively, such as 500 * 500 and 500 * 13, to calculate the final result.

💡Addition

'Addition' is the process of combining numbers to find their total or sum. The script concludes by adding the results of multiplications to find the final answer to 2^18 - 1.

💡Calculation

The 'calculation' refers to the act of computing or reckoning, especially by mathematical methods. The entire script is a detailed calculation process to solve the given mathematical problem.

💡Result

The 'result' is the outcome or answer obtained after performing a calculation. The script's final result for 2^18 - 1 is 262,143, which is the culmination of all the steps and calculations.

💡Algebraic manipulation

Algebraic manipulation involves transforming algebraic expressions using various mathematical rules and properties. The script demonstrates this by breaking down and simplifying the expression 2^18 - 1 using exponentiation rules and identities.

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