No calculator! Math Olympiad| Exponential equation.

Hi, let's quickly solve this using our calculator. All we need to do is simplify this number. and to do that, this number is the same as the square root of one one one one multiplied by one two three four five six one two three four five six plus two two two two two five right two five perfect now this is equal to the square root of one one one one multiplied by 10 to the power of six plus two two two two two zero plus five okay so we continue this same thing as square root of one million one hundred ten thousand one hundred ten multiplied by 10 to the power of six plus two hundred twenty-two thousand two hundred twenty multiplied by 10 Right plus five, all right, so we keep simplifying, this is equal to the square root of one one one. one multiplied by 10 to the power of six plus two brackets one one one one one right because two multiplied by this will still give us this multiplied by saying plus five, all right now we continue, so this same thing as was square roots of this one one one one is the same thing as nine million nine hundred ninety-nine thousand nine hundred ninety-nine divided by nine, right? All right, multiplied by 10 to the power of six. plus two brackets the same thing here we have nine nine five of them divided by nine multiplied 10 to the power of one plus five; now this is equal to what the square root of this nine nine Nine nine, that is nine thousand nine hundred and ninety-nine, is the same thing as ten thousand. minus one right ten thousand minus one divided by nine okay everything multiplied by 10 to the six to the power of two, plus two—now this sentence has ninety-nine thousand nine hundred and ninety nine, which is equal to one hundred thousand minus four to one, so it's going to be one hundred thousand one two three minus one divided by nine multiplied by 10 to the power of one plus five, all right, now we continue; this is equal to the square root of now 10,000 is sent in as 10 four to the power of four minus one divided by nine, then multiplied by 10 to the power of six plus two brackets this sentence 10 to the power of five minus one divided by nine multiplied by 10 to the power of 1 plus 5; now this is equal to the square root of; now let's open this bracket 10 to the power of four multiplied by 10 to the power of six 10 to the power of four multiplied 10 to the power of six is the same thing as when you have the same base and they are multiplying You take one base which is 10, then you add the powers, okay, to give you 10 to the power of 10. so let's apply it here, let's apply it here, so 10 to the power of four multiplied by 10 to the sixth power will give us 10 to the power of 10 minus 10 to the power of 6 times 1 is 10 to six divided by nine, okay, plus now here we have two brackets: 10 to the power of one multiplied by 10 to the power of five is sent in as 10 to the power of six minus 10 to the power one times one is 10 to the power of one divided by nine plus now we have this to be divided by nine divided by nine, let's get this to be the same thing as something over nine, which is what nine times five over nine right and nine times five we give us 45 over nine perfect so Let's continue from this side of the board, so but before that, if this is the first time you're If you come across this channel, please don't hesitate to click the subscription button and on your notification bell so I'll be notified anytime I upload a new video. Thank you very much. So this What we have is equivalent to what we have, which is equal to. So we have, let me recopy what we have. so that we continue, so we have 10 to the power of 10 minus 10 to the power of 6 divided by nine plus two brackets 10 to the power of six minus 10 to the power of one over nine plus 45 over Nine. All right, now notice that they all have the same denominator, so we can do what? Put everything together. Over nine, then we bring out what we have here. So what we have here is 10 to the power of 10. minus 10 to the power of 6 plus now this sentence has 2 times what 2 multiplied by 10 to the power 6 minus 2 times 10, which is 20, right, plus 45. Okay, so we have the square root of 10. of 10, okay, now 10 to the power of 6 plus 2 times 10 to the power of 6, so we have 2 times 10 to the 6 to the power of 6 minus 10 to the power of 6, 1, 10 to the power of 6 will remain, right? So we have plus 10 to the power of 6 remaining, then 45 minus 20 is what, 25 divided by 9, okay, so this is what we now notice that the number has decreased; this is now equal to the square root of This is the same thing as 10 to the power of 5 multiplied by 2, right, plus 10 to the power of 5 plus 1, right, because this is the same thing as 10 to the power of 5 plus 1 plus 25 is the same thing as 5 squared divided by 9, we are getting to the solution, okay. All right, so um, the solution continues. Now this is it, so let's simplify this further—this same thing as the square root of—so this same something like 10 to the power of 5 squared, right, since a to the power of m multiplied by n is the same thing Okay, or the other way around, so we can write this as this plus. Now this is the same thing as 10. 8 to the power of m plus n multiplied by 10 to the power of 1 right when you have 8 to the power of m plus n is equal to 8 raised to the power of m multiplied by 8 raised to the power of n, okay, so that's why I had to split it this way plus 5 squared, everything divided by 9. All right, now this is equal to the square the square root of 10 to the fifth power squared plus 10 to the fifth power multiplied by 10 is the same as 2 times 5 right plus 5 squared divided by 9 now I've formed what I've been trying to form all this way, and what is that? When you look at this, you notice that we have a plus b squared equal to a squared plus 2ab plus b squared; in this case, our a is 10 to the power of 5, that is our a Our b is what our b is, 5, so when you substitute this into this, you are going to get the whole of That's correct; that means we can replace the numerator with 10 to the power of 5. plus 5 right all squared, so let's go ahead and do that. So this same thing as square root of 10. to the power of 5 plus 5 squared divided by what 9 okay all right now this is equal to the square root of 10 to the power of 5 plus 5 squared divided by 3 squared okay so here we have what 10 to the power of 5 plus 5 divided by 3 all squared since both of them are sharing the Same power, so we can bring the power outside and put it inside the bracket. Now we have a square. sign and square so that both of them cancel each other, meaning that what we have here right now Now is what is equal to moderne the square root of 10 to the power of 5 plus 5 divided by sorry without The square root sign: we have 10 to the power of 5 plus 5 divided by 3. Now let's continue from here. We're actually almost at the end of it all now. What is 10 to the power of 5? That's 100,000, right? So 100,000 plus 5 is 100,005, so what we have now is 100,000 and what is 5 divided by 3, okay, so this is equal to when 3 divides 10, we have 3, remainder 1; if you divide 10, we have 3, remainder 1 When we divide 10, we get 3, remainder 1. Dividing another 10, we get 3, remainder 1. If we divide 15, we get Have 5, and this becomes our solution. Finally, thank you so much for watching, and have a lovely day.