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Strange Operators


The next topic in advanced algebra is strange operators. What do we mean by strange operators? Well, think about it this way, suppose you saw a question such as this. 3 and then some symbol 4. I don't even know what to call that symbol. Let's call it 3 star 4.

3 star 4 equals what? Obviously we have no way to answer this because we have no idea what that strange symbol means. The test absolutely cannot give you a strange symbol and expect you to know what it means by that symbol. So that would be a totally unfair, illegitimate question.

You will not see that on the test. But, the test could define a new symbol by giving us this rule. For all members x and y, let x star y equal x squared times y. So in other words, what we're gonna do, according to this rule, we're gonna take the first number, and square it, and then multiply it by the second number. That's what the rule is.

Okay, now that we have a rule, they can ask us to follow the rule. Then what is the value of 3 star 4? Now this is a totally legitimate question, because they tell us what they mean by that symbol, and then they ask us to follow that rule. So 3 star 4, we would just follow the rule given. We're gonna square the first number and multiply by the second number.

So that would be 3 squared times 4. Which is 9 times 4, which is 36. So 3 star 4 has a value of 36 when we're following this particular rule. Sometimes the test will give you a problem like this. They'll give you a problem with a strange operator. Some symbol that neither you nor anyone else taking the test has ever used as a math symbol before.

So it's very important to realize, every single person seeing the problem is gonna see this for the first time. They are encountering this particular symbol used in this particular way. For the very first time. The test will always give you the rule for following this symbol, and usually it's a relatively simple rule, and all you have to do is follow this rule by plugging some values in.

You see, the test loves this sort of question because most test takers panic. Most people mistakenly think, oh no, my math teacher probably covered this and I forgot it. And they go into all kinds of modes of panic and shut down. When you see this kind of question, don't panic. No one has ever seen this symbol before, and the rule for the symbol will always be there for you.

So just because it's new, don't think it's new just to you. It's gonna be new to absolutely everybody. Every single person seeing this question will be seeing this symbol for the first time. And so you're all starting on a level playing field. You just have to keep a level head and follow the rule that they give you.

Here's a practice question. For positive numbers a and b, let a, and I'm not sure what to call this symbol, let's just say arrow. Let a arrow b equal a over a plus b. What does p arrow p equal? So, pause the video here, and then we'll talk about this.

Okay, so this rule says, that the first number is gonna go into the numerator, and the sum of the two numbers is gonna go into the denominator. So p arrow p, that means we're gonna get a p in the numerator, and a p plus p in the denominator. Of course p plus p is 2p.

And we have p over 2p. The p's cancel, and we just get one-half. So p arrow p has an output, or a value, of one half. Here's another practice question. This symbol here, it turns out is a, is a Greek letter. The Greek letter phi.

For positive numbers p and q, let p phi q equal p plus 1 over q. So it's gonna to be the first number plus the reciprocal of the second number. What does, quantity 2 phi 2, 1 phi 2 and then phi 3, that's a hard thing to say. What does that equal? So I'm gonna to say pause the video here and then we'll talk about how to evaluate this.

Okay, so the first thing were gonna do. Is going to value what's in parenthesis. Order of operations tells us that parenthesis take priority. So, let's just focus on that one little piece in parenthesis. One phi two.

So one phi two, of course that's going to be the first number plus the reciprocal of the second number. 1 plus one half, which is 3 halves, I'll leave it as the improper fraction. Okay, now we can replace that whole parentheses with its value. That parenthesis, everything inside the parenthesis has a value of 3 halves. ous.

So we can just replace that. So one phi two becomes just 3 halves and we get 3 halves phi 3. So this would be the first number plus the reciprocal of the second number. 3 halves plus one-third, we find a common denominator. And we get an output of 11, 6, and that's the value. Here's another practice question.

Pause the video and then we'll talk about this. So of course, the heart symbol, that's a symbol we all know the heart symbol. But of course, we, we never used it as a mathematical symbol before. So heart as a mathematical operation is new for everyone seeing this question. So we're gonna to say let a heart b equal 2a squared plus b.

So the first number gets squared and multiplied by 2, and then we just add the second number. What does 1 heart 2 heart 3 equal? Well, first thing we're gonna do is set the parentheses, we're just gonna treat that separately, one heart two. 1 heart 2, what is that?

Well, that's gonna be 2 times 1 squared plus 2. 2 plus 2 is 4. So that whole parentheses just equals 4. So we can replace the parentheses with 4, then it's 4 heart 3. Well, this will be 2 times 4 squared plus 3. 2 times 16 plus 3.

Or 32 plus 3 which is 35. So that whole expression has a value of 35. Notice that all the operations here, these are just basic arithmetic and algebra operations. Once you follow the rule, it's always very easy to follow. Once again, just remember not to panic.

In summary, the test will give you strange operators that you and everyone else has not seen before. Don't panic, it's new for everyone. The test has to give the rule for the operator. It will always give you that rule.

Simply follow that rule. Remember to find the numerical value of the expressions inside parentheses first. You can replace the parentheses with the overall value of that expression

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