Birthday Card Math – Don’t Worry, YOU Got This!

Hello my lovelies, it's Susanna and today we're going to figure out when life begins. Because according to this birthday card, life begins at this age. The rest was just training. So let's take our beautiful expression here and let's figure out when life actually begins. Okay, there is a lot of stuff going on. We have all of this here in this square root, then we have these parentheses, all of this raised to the power of one over two, and then all of this raised to the power of four. So maybe we start with the exponents. There is a rule if you have something raised to the power of a and then you take all of this again and raise it to the power of b, then you are allowed to just multiply the exponents. So you take your base, your x, and multiply a by b, so you have a times b as your new exponent. And we would do this with the one over two and the four. So to find our new exponent, we are allowed to multiply one over two by four, which gives us four over two equals two. So in total we have one times two, which just equals two. So instead of these two exponents, we can just write a raised to the power of two here, so that our expression looks like this now, where we only have these parentheses and all of this raised to the power of two. But this is perfect because in here we have the square root and if we square a square root, then the square cancels out the square root and only the things that are in my square root are left then. So my expression looks like this now. Still a lot of stuff in here. Maybe we start with the logarithm here first. ln of one is always just equal to zero. So here we have zero plus this expression. So the zero, we can just erase this here. Then we have one problem less. What's next? Maybe here, nine times two, exclamation mark. What is the exclamation mark? This is the factorial. So we have two factorial here. How can we calculate it? Well, if you have something like five factorial, for example, this means that you always start at this number, at the five, and you multiply this by the next smaller number, by four, you multiply it by the next smaller number, next smaller number, until you get down to the one. So five factorial is five times four times three times two times one. In our case, we only have two factorial. So this means we start at this number, the two, and we multiply it by the next smaller number, by the one. We already arrived by the one, so this is the end. Two times one equals two. So two factorial is just equal to two. So in total here we have nine times two, which equals 18. Then maybe we can take a look at this thing here. This is not a fraction. There is not this line here in between these two numbers. This actually is a binomial coefficient. Five choose three. How can we calculate this thing? Well, there is a formula. If you have n choose k in general, then this is calculated by a fraction. You take the first number, the n, and use the factorial. And in the denominator, you take the other number, the k, and also factorial. And then you multiply this by n minus k. So you subtract these two numbers. We have n minus k, and then again factorial. So in our case, we have five choose three. What does the formula say then? You take the first number, n factorial. So in our case, five factorial. Then you have k factorial. So in our case, three factorial. You multiply it by n minus k. So five minus three. And then factorial again. Five minus three equals two. So we have two factorial here. So I erase this already. Two factorial. And then this looks like this. Let's calculate this thing. Five factorial. We've already seen what it is. We start at the five and multiply it down to the one. Three factorial. Same thing. But we start at the three. Three times two times one. And two factorial. We already had that. We start at the two and times one. Now we can cancel some things out here. We have times one. Okay, it doesn't matter at all. We have the two here. The three here. We can reduce four over two. Which equals two. So in total, we only have five times two. Which gives us ten. So this thing here equals ten. And my expression. I wrote it down on the next page again. I have my 10 here. My 18 here. This thing is not there anymore. So my expression looks like this now. It's getting better. We will find out when life begins soon. Let's take the sine of pi over two next. How can I calculate this? I first want to find the value of this green thing. And after I found that, then I'm going to calculate the square root of it. But first this in here. Either you know this value by heart or if you don't, you can always take a look at the unit circle to find the values for sine or cosine. The unit circle is a circle of radius one. So I have the value one here, one here, negative one here, and negative one here as well. On the x-axis, I can find the values for cosine. And on the y-axis, I can find the values for sine that I'm interested in here. Then I have to take a look at the angle. I'm interested in the angle of pi over two. Where do I find this value now on my circle? I always start here. And if I would go a full circle, then I would have an angle of two pi. We are not interested in two pi. We are interested in pi over two. So half a circle would be half of this. This would just be pi. So if I'm here, this would be the angle of pi. We are not interested in pi. We are interested in pi over two. So half of pi. So we don't go a half circle, but half of it. So we only go a quarter circle and stop here. This is my angle pi over two. And at this point here, I want to find the sine of pi over two. So I take a look at the y-axis because I can find the values of sine here. And which y value do I have at this point? It is one. So sine of pi over two is just equal to one. Then I need the square root of one, which in total is just one as well. So I have all of this over one. So the denominator, this over one, I don't need this at all. So I don't have a fraction here actually. I can calculate these two numbers then. I have 10 plus 18, which in total equals 28. So this is the only thing that is left. And what is this thing here? This is a matrix. In here, we have three columns and three rows. So a three by three matrix. And these lines here tell us that we have to find the determinant of this matrix. How can we find the determinant of a three by three matrix? For example, with Sarrou's rule. Sarrou's is the name of the French guy. So actually the pronunciation would be Sarrou, but never mind, we call him Sarrou. And the rule says, take the first two columns and write them down here at the end again. So this first column, this is the second column. And now we have these diagonals. Number one, diagonal number two, diagonal number three. This should be the four. So let's make it like this. And we have the other orange diagonals here, here, and here. And to find the determinant, we now only have to multiply along our diagonal. So we have three times two times one, which equals six. And then we add the same thing with the next diagonal. Zero times negative one times zero equals zero. We add the same thing with the last green diagonal. One times one times four equals four. And then we always subtract the same things with the orange ones. Zero times two times one equals zero. We subtract the next orange one. Four times negative one times three equals negative 12. And we subtract the last orange one. One times one times zero equals zero. In total, we have six plus four, which equals 10. Negative negative 12 is positive 12. So in total, we have 22 as a result for the determinant of this matrix. 28 plus 22 equals 50. So life begins at 50. The rest was just training. Well, some of us are still in the training. Others might live a wonderful life already. Either way, I wish all of you a wonderful day. And I hope to see you in one of my next videos. Take care.