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How to Solve Radical Equations Step-by-Step | Full Lesson + Examples
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May 1, 2025
In today's lesson, we dive into solving radical equations — an essential skill for algebra and beyond! I break the topic into simple, easy-to-follow parts: ✅ Isolating the radical ✅ Squaring both sides correctly ✅ Solving for the variable ✅ Always double-checking your solution (so you don’t fall for extraneous answers!) We cover examples including: Radical equations with one radical Equations with radicals on both sides Solving cubic (third-root) radical equations Fourth-root radical equations You'll see me walk through each problem step-by-step, explaining the logic behind every move — including what to watch out for when checking your solutions. Perfect for beginners or anyone needing a refresher!
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all right so we are starting we're going
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to learn how to solve radical equations
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and inequalities so today I'm going to
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break this up into two sections
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subsection the first one is we just
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going to talk about radical equations
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simple and then next tomorrow on
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Wednesday we're going to talk about
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inequalities with radicals right so here
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we we're trying to solve equation that
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have radicals in them okay that's what
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we're going to do this i gave you two
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three main uh wrinkles here the first
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one very simple one and the second one
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you have two radicals on either side of
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the equation and the last one we're
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going to talk about a cubic equation and
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how do you solve these kind of equations
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okay so now what I have here is I want
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to solve this for x okay I have solved
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roo of x + 2 + 4 = 7 and I want to
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solve it for x so what comes into mind
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what what would you want to do if you
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had never done this before this is your
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first time seeing this what would be
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your first logical thing to do yeah
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subtract the four subtract the four
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right subtract the four so now we have
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square root of x + 2 right
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is equal to 3 now what is the next thing
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that we are trying to achieve here we
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want to solve for what what are we
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solving for really uh x right but
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there's an obstacle what's the obstacle
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the obstacle is the radical
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right so how did you get rid of the
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radical what did you do here you square
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root square both sides we square both
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side right so we're going to square both
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side
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and now once you square both side what
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happens to the radical here it
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disappears it's
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gone so when you square both side the
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radical is gone because the square root
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is x + 2 to the 1/2 so squaring it means
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you take away the radical so now you
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know if you what x + 2 = what
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what's 3 square 9 right nine and then
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we're
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subtracting so we get x is seven right
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so seven is our answer but what you find
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is sometimes the answer can be extra it
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could be a wrong answer so what we do we
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double check we're going to check it
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right so I'm going to go back to the
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original problem i'm going to replace x
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by seven to see if this equation will
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verify it right if it holds so if I go
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back here
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no no no you can't
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right so I'm going to replace x by seven
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in the equation so that's going to give
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me seven right plus
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2 + 4 is equal to 7 let's double check
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this statement right 7 + 2 is 9 so this
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is square root of 9 + 4 and I want it to
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be equal to 7 this is what I'm trying to
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aim for right of 9 is 3 so 3 + 4 is 7 7
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is equal to 7 so therefore this is good
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right so therefore the x= 7 is a good
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solution so you always want to double
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check your answer when you cancel
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radical to make sure you don't have the
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wrong answer right so first thing again
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is you isolate your
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radical you isolate it next we get rid
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of it by raising it to the power of two
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because x + 2 radical is the same as
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saying x + 2 to the 1/2 and to get rid
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of the 12 you just raise to the power of
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two so that that takes care of the
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radity okay any questions on that are we
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good on that right all right so we're
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going to move on to the next
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set can I erase this here
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all right
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mhm
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all right now here what do we have what
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do you notice
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here what do you notice we have two x's
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you have two x's two radicals right on
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either side so what do we do we're going
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to proceed the same way i'm going to
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square this right
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and I'm also going to square this but
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now there's going to be a problem on
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this side here i can't just see the
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radical does not cover the two it only
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covers what the x right now on the left
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side we're good cuz because the radical
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covers everything we're left with x - 12
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that is fine right on the left side
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we're going to be okay because the
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radical covers everything so if you
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square it it takes care of the radical
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now here we're going to have to foil
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this so that's going to give us this so
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we have a double work to do here right
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so now I'm going to go step by step here
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i'm going to have to do it all the way
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right so this is x - 12 = 2 * 2 which is
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4 right 2 * x that is -2 x and then of x
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* 2 that is -2 x and then I have
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roo of x
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time of x that gives you what 2 x square
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of x² which is x right cuz I have two
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negatives so therefore it's just going
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to give me what plus x because I'm
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multiplying roo x times the square
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root of x that gives me x okay now watch
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this so I'm not going to combine like
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term right
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so that give me four these two can be
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combined because the out the the numbers
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that are under the right are the same so
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that gives me4 x plus x here right now
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what I'm going to do is I'm going to put
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this on this side and again I'm going to
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isolate my radical right cuz I still
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have another radical so I need to
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isolate it so I'm doing twice the job
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here right so I'm going to first
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subtract the x to this guy so that is
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gone and then I'm also going to subtract
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four here right so I get -16 is equal to
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-4 roo of x now I can square it
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here but I won't do that because I want
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to make sure my stuff look decent so I'm
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going to divide everything by what
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here4 right by4
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so I get four is equal to square root of
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x now what's the logical last step both
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you square both square both side right
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we square both side so now we get what
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16 is equal to x so x is equal to 16 and
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now what's our next step we're going to
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double check to make sure this is the
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right answer I'm going to take this and
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I'm going to plug it in here to see if
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this verifies right so I'm going to go
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back to the original and I'm going to
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replace x by 16 and see if this is going
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to hold Right so what's 16 - 12 that is
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4 right square of 4 is equal to 2 -
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4 square root of 16 here is four right
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so this is 2 = to -2 what do you say
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not not equal so what would you conclude
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about this problem
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no no solution right no solution now
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wait do we have to check it you have to
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check it if you don't check it cuz you
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may not hold you have to check the
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square root so you have to check it at
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the end to make sure that this is either
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a solution or this is not a solution so
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at the end you always want to check it
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okay here it did not work so therefore
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this is not a solution to the problem so
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this problem does not have any solution
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oh what is this there's something in her
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eye something in her eye are you looking
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that close to her eye
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all right so in this case you see that
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the solution is not work so we always
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must double check it right we must
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double check our answers we must double
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check our answers all right okay now the
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next problem I think you guys are going
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to be able to figure this out right even
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though this is a cubic function right i
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have 2 6 x - 3^ 1/3 - 4 so what would
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you want to do here Nathan what would be
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your first step
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uh add the four add four there you go we
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add four first and then we get two right
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and then 6x - 3 to the 1/3 equals to
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four then what you distributed the two
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divide you mean divided by two right
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divide by two all right and I have 6 x -
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3 to the 1/3 = 2 now what will be the
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next logical step
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here this is 5 to 1/3 so how do I get
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rid of that that exponent to the 1/3
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cube it right because it's a one to get
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rid of one third you going to raise it
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to the power of what
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right there you go
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so we going to cube it and then we going
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to cube this side because this is 1/3 so
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if you cube it it gets you get rid of
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the exponent right so now this is gone
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so I'm left with what 6x - 3 = to what 2
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to the 3 right cuz if you do one thing
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on one side you must repeat it on the
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other side okay so you have 2x - 3 = to
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2/3 23 power is 8 so this is 6x - 3 = to
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8 you add
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3 right 6x = 11 and x is equal to 116
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right
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116 all
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right again we can double check the
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answer let's double check it i'm going
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to go back to the original problem and
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I'm going to replace x by 116 right i'm
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going to double check it so I'm going to
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go 2 6 * 11 / 6 - 3 over 1/3 - 4 is
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equal to 0 so let's double check that
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right so these sixes are going to cancel
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they're gone so I have 2 11 - 3 - 4 is
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equal to 0 i'm double checking so I'm
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putting the question mark oh yeah 1/3
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thank you very much so 11 - 3 is 8 right
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so it's 2 * 8 1/3 - 4 is equal to 0 a to
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the 1/3 that is two right 8 to the 1/3
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is two so this 2 * 2 - 4 is equal to
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zero yes that is true so therefore this
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is a solution this is a good solution
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okay so in this problem all we doing is
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just solving double checking solving
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double checking the answer is to make
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sure that we are doing it right okay all
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right there's one more thing that I want
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to do here and then tomorrow we're going
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to talk about inequalities now I was
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just talking uh radicals
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you good
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all right so I want you all to see if
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you can figure this one out here right
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yeah
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so we have uh three right parentheses
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the fourth root
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of 2x + 6
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right - 6 is equal to zero all right um
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who wants to help me out here go ahead
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okay mhm plus six so three plus six all
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right and then you divide three from
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both sides all right divide three on
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both side so we divide it by three
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divide by three all right okay and then
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you get
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Mhm fourth root of 2x + 6 = to what two
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all right so do you raise it to the
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power of 1/4 no four 1/4 four right four
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right
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and then what happens to the radical uh
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it gets out of there gets out of there
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you got 2x + 6 = to 16 right yeah mhm
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and then you - 6 so it's 10 so x is five
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x is 2 x is 10 and X is five and then
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the last step will be to double check it
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plug it in here all right so 2 and 5
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that's 10 10 + 6 is 16 the fourth of 16
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is 2 right 2 * 3 is 6 6 - 6 is 0 okay
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you can do that that's just something
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you can do all right so this section is
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not really that complicated once you
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just apply the techniques that we
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learned all right
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yes sir oh sorry
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i apologize
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my bad
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my bad yeah
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uh yes tammy gosh
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yeah yeah why don't we
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replace there
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all right cuz you five here right we got
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three right 4 of 2 * 5 + 6 - 6 is equal
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to zero i put a question mark cuz we
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want to double check this right so that
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is
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three of 25 is 10 right 10 + 6 is
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16 - 6 = 0 right now what's the fourth
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root of 16
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it's two two right so be 3 * 2 - 6= 0 6
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- 6 is equal to 0 right true or false
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true right therefore
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five is a solution
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right find the solution because that
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works all right okay so what I did was
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uh so you can start on your uh you can
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start working on your homework now all
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right so 1 through 10 23- 27 for honors
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and for regular we got that okay