it came from her textbook!

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[00:00:00]solver. I can't even read her. 11th grade homework, huh? Yikes. How many years must you teach before your patience shrivels up enough to type this as homework and think, "Yeah, looks good." Well, you know, all teachers do have their quirks. I had a teacher who used to show us how to find used socks and underwear on eBay. And when I taught, I'd write up any student who walked into my classroom without a limited edition calculator. I think that was one of the most important ways students could show they took math class seriously. One of the helicopter moms complained to me, "My husband and I take our child's education very seriously, and we'd appreciate it if his math teacher wasn't a deranged lunatic." And I said, "Mrs. Chapman, your son uses a TI 84 Plus." Anyways, we understand a good student acclimates to the environment. And this student has done exactly that because their post isn't even about the hideous formatting.

[00:01:01]They're just asking for help on the math. He didn't even think to clarify what the hell is going on until people replied, "What? What the hell is this?

[00:01:12]Solve this? I can't even read it. This is not well-defined notation. It's nonsense. Your notation is [ __ ] little bit aggro. I don't think OP wrote this problem, so there's no need to be mean about their dumb [ __ ] diarrhea dookie notation. Just use solve. Duh. The button next to the cerebral cortex.

[00:01:33]Yeah. All right. I suppose we ought to solve it. Then I want to show you a math problem I saw a couple years ago that looks even worse before we finish with a bit more notation talk. To solve this, we could begin by checking an obvious candidate solution. 6 is 3 * 2. And on the left side of this equation, we have a factor of three. If we force it to be just that, 3 by setting x equal to 1, we'll see everything actually works. If x= 1, we have 3 to the power of 1 ultiplied by 8 to the power of 1 / 1 + 2. That's 8 to the power of 1/3. That's the cube root of 8, which of course is equal to two. The left side will equal the right side. And so, God indeed x= 1 is a solution. But no good math student can stop there because what if there is another solution? In fact, we can be sure there is another solution just by examining the expression and thinking about the graph. 3 to the x is simple.

[00:02:43]For large negative powers, it will be close to zero. At x= 0, it will be 1.

[00:02:49]And then it grows exponentially. And certainly, this all by itself would equal 6 at only one point. But look at the whole expression. This factor used to be a simple exponential like 3 to the x, but then it took an x to the denominator. And that's going to introduce some unique behavior near -2 where the denominator equals 0. When x is near but less than -2, the denominator is very small and negative, meaning the exponent will be a massive positive number causing the whole function to shoot up. When x is near but greater than -2, the exponent is a massive negative number. So the whole thing crashes back down to zero before going through the exponential growth again due to the 3 to the x factor.

[00:03:42]Hence there will clearly be two solutions where this expression equals 6 and the equation is true. One is at x= 1 essentially due to the 3 to the x factor. But how do we find the other one that's popped up due to the 8 to the x overx + 2? One option is to use logarithms and algebraic brute force.

[00:04:05]Logarithms for the uninitiated undo exponentiation. There's a log function for every positive base other than one.

[00:04:13]And each function will output the exponent its base would need to equal the input. Wielding this tool like a hammer on this problem, we end up with a pretty nasty quadratic. Yeah, not even proper notation can save the looks of this one, but we can just use a quadratic formula and wash our hands of it. B plus or minus the roo<unk> of B square - 4 A C all over 2 A. And there we go. Here on the top, theB plus solution is indeed equal to 1. And theB minus solution, this glorious specimen is about 3.26.

[00:04:52]Our graph was a super rough sketch, but certainly these solutions don't raise any red flags. That makes sense. This brutish solution leaves one feeling a touch hypocritical. However, if we are to cast stones at this grotesque notation, then shouldn't we offer something better than a grotesque solution? Thinking back to when we factored 6 is 3 * 2, a more elegant method requiring no quadratic formula is within reach. Since 8 is itself a power of two, we can make this left side match up even better with the factorization on the right by writing 8 as a power of two. 2 ^ of 3. Doing that, this whole term will become 2 ^ of 3x over x + 2.

[00:05:40]And if we continue to massage the equation in this factored form, both solutions will pop out. Now of course we can divide by three and divide by two quite cleanly making the right side equal to one and on the left each exponent gets reduced by one. In 2's exponent subtracting 1 is of course the same as subtracting x + 2 from the numerator which can be written in this factored form that will come in handy later. Finally to get the variable out of the exponents we will need to take logarithms. We could do log base 3 or log base 2. But things are a little nicer if we do log base 3. So we'll take log base 3 on both sides. Now the log of this product is equal to the sum of the individual logs. And log base 3 of 3 to the x -1. Well, the bases will cancel out and that's just going to be x -1.

[00:06:33]Then we'll have to add log base 3 of this factor. And by our log properties, we can pull that exponent out as a factor. So in front as a factor, we'll have the exponent 2 * x -1 over x + 2.

[00:06:49]And on the right, 3 to what power produces 1? Of course, it's 0ero. Having zero on this side is really useful because it allows us to use the zero product property. If only we can factor this side of the equation. And of course we can because each term has an xus1.

[00:07:08]So let's pull that x -1 out of both terms. Thus we factored it as x -1 * 1 + 2 / x + 2 * log base 3 of 2 and this product equals 0. Of course that means this factor equals 0 giving us the x = 1 solution or this factor equals 0. If this factor equals 0, let's subtract this log term to the right side and then multiply everything by x + 2. That's going to leave 1 * x + 2 on the left side. So x + 2. And on the right side, we'll have the part we subtracted. So - 2 log base 3 of 2. Finally, subtract two from both sides. And there's a nice factorization if we like. And we can write the answer as -2 ultiplied by log base 3 of 2 + 1. Or if you'd prefer it written as a single term, 1 is the same as log base 3 of 3. So then we could combine these logs by using the log product rule. And there we have our non- integer solution, but in a much cleaner form. Yeah, this monstrosity and this are equal. It's no surprise that the more elegant method produces the same correct solution, but in a much more pleasant form. Well, there you go.

[00:08:26]That's how to solve this nasty 11th grade math problem. Now, are you ready for a math problem whose notation is even more hideous than this? Think about your typical viral PEMDAS problem. They always look a little ugly because they always use the obelis, but it's absolutely nothing compared to this.

[00:08:58]If you've seen a problem more viscerally repugnant than this one, let me know. Or maybe don't. The original poster of this deplorable exercise said he got 3225ths, but his friend says the answer is two.

[00:09:13]But if his friend sent this to him, I wouldn't trust that friend to ascertain the value of a $1 bill, let alone the value of a horrifically written equation with a question mark on the right side.

[00:09:25]Indeed, in terms of determining the value of this equation, we can't. We don't know how to do it. So, I will just content myself with finding the value of the question mark. But that of course will depend on which operations we choose to give priority. So, what do we need to figure out to solve this? Well, firstly, of is a term that means multiplication. A lot of the people who read this problem on the original thread seem to be unfamiliar with that, but it's kind of common. What is 1 next to 1/4? I'm assuming that's just a mixed number, as in 1 + 1/4. I'm absolutely not going to entertain the possibility that it's a multiplication by juxtaposition because honestly, give me a break. That leaves two ambiguous items to consider. How many copies of 1.25 are there? Are there 2/3 of 1.25 or 56 divided by 2/3 of 1.25? And the other point of ambiguity, what the hell is this? Is it 2/3* the fraction or 2/3* the denominator? The alignment here is repugnant. You know, honestly, I can't bear to look at this another moment.

[00:10:39]Here are the solutions. What a abysmal problem. But honestly, while this notation disturbed me at first, it's not that bad. I think it looks so horrible for a couple of reasons. The fraction in 8 exponent has this characteristic that computer fraction notations often have, which is a lot of empty space here around the vinkulum. This causes the exponent to seem quite misaligned compared to 3's exponent. With proper type setting, it looks like this. The x exponent is aligned with the center of the fraction exponent. But here, the x + 2 is way lower than x and the rest of the exponent is way higher. It's gross.

[00:11:25]And worst of all, it makes one wonder if the intended expression is this, or if because that x is so high, it's meant to be this. And of course, what about the multiplication symbol? A dot on the bottom. Using an x as a variable does not give you the right to use a decimal point for multiplication. But then, what's the alternative? Based on how this expression is written, we certainly couldn't expect the author to bust out an actual multiplication symbol, the salt tire, to contrast with an actual variable X. This is a 10point computer modern math axe, by the way, a font created by Donald Kuth. It's a font that would make anybody shiver in ecstasy. I assumed the teacher must have lazily typed this equation using some word processor, but I'm not sure how they did it. If I use Google Docs, I can get this. If I use Word, I get this. The exponent alignments certainly vary, but none of them look as bad as this. So, what did the teacher use? Well, I would suppose the key word is that whatever they did, they did it lazily. But guess what? According to OP, it's not the work of a single teacher. It's literally from the textbook. How do you get this abomination in the textbook? I don't know. But however they formatted it, surely an asterisk, a five or six-pointed star, would have been an easy alternative that isn't without precedent. In 1659, Swiss mathematician Johan Ron used the asterisk to denote multiplication in his tocha algebra. In 1668, Thomas Branker translated the work originally written in German to English and kept the asterisk notation. In 1692, the asterisk was used for multiplication in volume 17 of the journal Philosophical Transactions. This is one article from that volume, though I can't find the article featuring the asterisk myself, so I'm relying on Kajori's account. Of course, the dot itself is a nice popular multiplication symbol, but it's supposed to be in the center.

[00:13:35]Although we do get close to a bottom dot for multiplication on page 156 of a 1710 publication by the Berlin Academy. They say that multiplication is to be denoted by a dot or comma as seen here. A * b + c. Huh, that's pretty close. They also say that should another symbol for multiplication be required, the liitsian symbol, a sideways C, is preferable to the salt tire. And you know, since the decimal point is right next to the comma on the keyboard, maybe the author of the problem made a simple mistake. They were trying to follow the example set by the Berlin Academy in 1710, and they just made a typo. If we just clean both of these problems up to be more in line with some of those old school standards, I think they'd look a lot good.

[00:14:30]>> Riverside Drive held me up when the first man I love died when I could only say I love you through and say I miss you.

[00:14:43]Last time I wasn't talking shells out of the shotguns, someone with a much bigger heart