John didn't fully apply PEMDAS. He stated that there were no exponents. Roots are equivalent to fractional exponents, e.g., √n = n^0.5 or √n = n^(1/2). Simplifying the roots first is the most direct way to solve the problem: Starting with the numerator, √8 / √2 × √4 = 2 x √2 / √2 × 2 = 2 x 2 = 4 Dividing that by the denominator (2) gives the correct answer, 2.
You’re absolutely right that he missed doing the roots in the E of PEMDAS. And evaluating the numerator is even simpler than that because it’s all just multiplication and division: √8 / √2 × √4 = √(8/2×4) = √16 = 4
Strictly the root operator has a different priority to exponents and would be covered by the O in BODMAS but not the E in PEMDAS. PEMDAS is irrelevant to the solution to this and BODMAS almost so. What is not taught at school is that both BODMAS and PEMDAS are only a simplified version of the full order of operator precedence. In practice if the equation can be viewed as being ambiguous always use brackets even if the full set of operator precedence rules resolve the ambiguity as by far the vast majority of people do not know the full set, indeed the vast majority of mathematicians don't know all of it.
@@GuardOfGaia The O in BODMAS and the E in PEMDAS are exactly the same thing. BODMAS and PEMDAS are exactly the same thing. BODMAS, PEMDAS, BEDMAS, BIDMAS, GEMDAS, PEMA, BOPS, BIPS, GEMS, etc... There are lots of variations of the acronym but they're all just different ways of referring to the same calculation process. The second step of that process, referred to variously as "Exponents" and "Orders" and "Indices" in the acronyms, is to evaluate all the powers and roots in the expression.
@@gavindeane3670 the O is for other operators not just exponents so would cover the root operators, unary +, unary -, dot product, cross product etc. Admittedly at school we were taught it as Brackets, or, division/multiplication, addition/ subtraction so would be more Boolean operators than exponentiation. The important this is to understand there are a huge number of different operators that have specific precedence levels in addition to those commonly considered in BODMAS or PEMDAS and those have much more relevance to the posted equation. Several of those are mistakenly assumed by many to be the same priority as one of those which are included - much of it is because keyboards used for writing the symbols commonly do not include the correct symbols and others which are similar are used instead and have since entered common usage. Thins like ✓4 is not the same as 4 raised to the power of 1/2 in terms of operator precedence. Even with that both the ✓ and / symbols are still only symbols which approximate the appearance of the actual operator. The root operator √ should continue to the right horizontally and everything below that extension would constitute the operand, the / is a typographical approximation for the horizontal bar in a fraction - the operator for division is ÷
@@GuardOfGaia I don’t know where you got that from. The O in BODMAS stands for Orders. It is exactly synonymous with Exponents in PEMDAS and Indices in BIDMAS. Literally just different names for the same thing. Yes, of course there are more types of operation in mathematics than that, but remember that these acronyms are designed for children. The acronyms are used to introduce 10 year olds to the basic concept of precedence and so obviously the acronyms only cover the operations that children are using at that age. Some of the examples you’ve given under your “Other” heading are things that they won’t have learned at that age, so why would those operations be within the scope of the acronym? Moreover, you’re bundling a whole load of operations under the “Other” heading which don’t all have the same precedence as each other. We already have the problem that different letters in the acronym refer to different operations that have equal precedence (D&M, A&S). Now you’re proposing something even worse: a situation where one single letter would refer to a whole collection of operations (although you haven’t even given the complete set of operations that you would put under “Other” yet) that do NOT all have the same precedence as each other. That would completely defeat the point of the acronym. Have you got an example of a case that illustrates your point about √4 and 4 raised to the power of ½ having different precedence? You’re correct that √ isn’t the proper radical symbol, but it’s often what we’re limited to in basic text editing. For a single value like √4 it’s OK though, and you can always use brackets to get round the problem for a more complex expression. Here in a RU-vid comment I can easily write √(1+3) where in proper mathematical notation I would extend the horizontal line of the radical symbol over the 1+3 and not need the brackets. I don’t know who has been talking to you about division though, but you need to stop listening to them because they don’t know what they’re talking about. The / symbol is absolutely NOT an approximation for a horizontal fraction bar in the way that the typographical √ character is an approximation for the proper radical symbol. The / symbol is a proper, correct, mathematical symbol. It is the operator for division. The ÷ symbol is deprecated and should not be used. Nobody who uses mathematics to actually do things uses the ÷ symbol for the inline division operator. A horizontal fraction bar is drawn as long or as short as it needs to be for the particular context. Its length communicates exactly what is and is not part of its numerator and denominator (see, for example, the expression at the start of this video). The / division operator is not capable of communicating that. People (including the guy who makes the videos on this channel) can get themselves into right a mess by thinking of / as the inline representation of a fraction bar instead of thinking of it as what it actually is: the division operator. They end up making all sorts of strange assumptions about how the / symbol can group what is and is not in its numerator and denominator, instead of simply looking at the precedence the division operator relative to the precedence of other operators around it.
I got this in about thirty seconds doing it in my head, I was pleased with that because although I would have known how to do it even a few weeks ago, before I started watching these short tutorials has increased my confidence, it is after all 55 years since I left school and although I studied math to a much higher level for my engineering qualifications, the work I subsequently did as an engineer didn't require me to do much mathematics. With lack of practice, especially since I retired in 2009 confidence wains. It is also very interesting to watch John's lessons to see how he teaches, every lesson goes right back to basics. Some criticize him for that saying he is long winded, but those people don't think of others who might be new to the channel, new to math or like me a long time since I studied math. Indeed, there is nothing better than repetition to reinforce memory.
the only problem I have with him is he is too kind in some of his grading. There are teachers out there who will NOT give partial credit like he does, and they expect all the work to be shown and the answer to be fully simplified. A bit more advanced and hard to show without a black board or scrap paper, but watch some more videos and you may see what I mean. Otherwise this guy is the goat.
Given that this is a twenty minute video, there are two learning points that are surprisingly absent (plus there's a straight up error in the application of PEMDAS). There's a missed opportunity to explain how GEMDAS is superior to PEMDAS, with G for Groupings. You talk about treating the numerator as its own separate problem, but you missed the opportunity to explain the reason for that: the vinculum (the horizontal fraction line) is a grouping symbol just like parentheses are a grouping symbol. The part of the expression that is grouped by the vinculum, in this case the √8 / √2 × √4, is evaluated in the G step in GEMDAS. Also a missed opportunity to point out that since it's all division and multiplication in the numerator, you can replace √8 / √2 × √4 with √(8/2×4) You got part way there by doing √8 / √2 = √(8/2) and then √4 × √4 = √(4×4) but why not go the final step and point out that the whole numerator is even simpler than that because it all goes under one radical sign. Finally, there absolutely IS work to do in the E of PEMDAS or GEMDAS here. In the E step you do powers AND ROOTS.
Square it, yielding (8÷2*4)/4 or 8÷2 - but that's the square of the answer, which is then 2. Or, factor the numerator to 2*sqrt(2)÷sqrt(2)*2, cancel out the sqrt(2) to end up with 2*2. Cancel a two with the denominator, answer is still 2.
A The square root of eight is just shy of 3. The square root of 2 is just shy of 1.4. The square root of 4 is 2. 2x1.4 =2.8.just shy of 3 divided by 2.8 is about 1. 1/2 is the answer. This problem required me to use pen and paper.
The answer is 2. You've solved it as if there were parentheses around the √2 × √4 part. You don't need any decimal approximations of the square roots to calculate the numerator: √8 / √2 × √4 = √(8/2×4) = √16 = 4
I can see what you did there. You're subtly trolling the root 2 cubed = 2 x root 2 from Terrence Howard's BS loop and applying it to a maths problem. Bravo!
It's way to much time consuming I barley took 10 min to solve this Just divide root 8 by root 2 which in root 4 and root 4 × root 4 is 4 Now u get 4/2 which can be simplified to 2/1=2
Why do you talk so much? Get to the point of solving the problem. Maybe teachers like you are making it too complicated. You talk way to much! I taught math and would never talk so much. My kids would have lost interest and zoned out! Talk less!!
what did Aunt Sally do? At an elementary school where I used to teach, she very guilty of fashion crimes... our math teachers dressed very badly that week. Old women with NO fashion sense. :)
Explanation is so long so im grtting confused , perfer to jusst show how tonsolve it , and after that talk and show diff things ❤, bout thanks for the videos anyway😊
It’s not really necessary to know the properties of square roots to arrive at a reasonable guess… approximating values to the nearest 1/2 gives sqrt(8)~3 and sqrt(2)~1.5… so, 3/1.5=2, and sqrt(4)/2 cancels out, so the final answer at least approximates to 2… Therefore (b) is the probable answer.
Hmm, I was always taught to do the maths in order of the letters which means do the multiplication first then the division. This gives answer A. So why is this incorrect?
That's shockingly bad teaching. That's a common and understandable misconception. It's the biggest problem with these silly 6 letter acronyms. Even a simple example like 3-2+1 will show you that treating PEMDAS as 6 separate steps is incorrect.
super simple remembering that x/y is the same as x * (1/y). we can rewrite the equation to (√8 * 1 * √4) * 1 / (2√2) Numerator is √8 * 1 * √4 which equals √32 denominator is 2√2 Denominator is irrational so we multiply the top and bottom by the root. which is √2 in this example. Numerator is √32 * √2 which equals √64 which finally equals 8 denominator is 2 * 2 which equals 4 Final answer 8/4 which equals 2
The x/y = x * (1/y) identity is true and you have got the correct answer, but it seems a lot simpler to use a different mathematical identity. √a × √b = √(a×b) and √a / √b = √(a/b), so the numerator here is simply √8 / √2 × √4 = √(8/2×4) = √16 = 4 You only need to calculate one square root in this problem, and it's the square root of a perfect square.
Before watching. I’d go sqrt(8): sqrt(2)=sqrt(4)=2. Need to know your root laws, or you get stuck right here. Now sqrt(4)=2, so the nominator becomes 2x2. Divide by 2 and the answer is b. Now, the other way to go is sqrt(2)*sqrt(4)=sqrt(8). Aha, sqrt(8):sqrt(8)=1, so the answer would be a. I think the order of operations for the nominator here is left to right, so my ultimate answer is b. However, pemdas states m=multiplication, d=division, and m comes before d in pemdas… I still stick with b, but in my opinion, pemdas should get some extra acronym added to make this exact issue more clear. One way or the other 🤓 now I’ll watch and possibly get to eat my analysis of this one 🥸watching now. Pemdas does not state to work inside brackets out. That is another rule added on top. Aha, now comes the left to right. Another rule not obvious from the acronym. Now comes the internet chaos… So, I propose change pemdas to pioltremdas. Parentheses inside out, left to right, and the rest. Weird acronym, but still rememberable.
You raise a very good point about the acronym, but other people have already solved that problem rather more neatly 🙂. The acronym you're looking for is PEMA: Parentheses Exponents Multiplicative Additive
@@mr.mxyzptlks8391 I don’t see why the acronym would need to clarify any implicit left to right rule. The need for that clarification only arises when people are given misleading acronyms that seem to suggest a hierarchy between M&D and a hierarchy between A&S. If you don’t give them the misconception in the first place then you don’t need to say things to undo the misconception. This is a sentence. You did not read that as “sentence a is this”. You don’t read 4/2 as “two divided by four”. I can’t imagine there is any way that it would occur to someone to read 3-2+1 as 3-(2+1) except by them being given a silly 6 letter acronym that suggests a hierarchy between addition and subtraction. If you never create the problem, you don’t need to do things to fix the problem.
I got 4. Force of habit to just do that top stuff from left to right, instead of doing the multiplication first. Got the right answer after I fixed that. Would have still been wrong, I assumed the square root of 8 was 4, despite that not being possible. Explains why I never passed math.
