Ok kids, so today we learned, or were supposed to learn, about simplifying these annoying mathy thingys called radicals. You've probably heard about them before... if you were paying attention. Cool cool. Here's some fun stuff you might want to remember. Yay. √A X B = √A X √B we call the blue font "baby radicals" or "cube root baby radicals" if they are 3√ (how creative) A√B is called a mixed root (3√7 , 9√3 , etc). It is a non radical with a radical. √A is called an entire root (√67 , √23 , √36 , etc). It's a radical by itself. Positive or negative. How very complicated. So. You're given a radical number. How the heckity h*ck are you supposed to simplify that thing now?? First, factor it. Yep, we're doing factoring again. Which, I mean, that's great. If you love factoring. When you find a perfect square in your lil tree thing you got going on, stop. For example, you don't need to take 4 to 2 and 2, because 4 is a perfect square. At the end of your factoring, you should have mostly perfect squares left. If you don't, keep going. If you have a couple non perfect square numbers left, it's all good. We don't discriminate here. Factoring Tip: the bigger the perfect square you can find, the less work it is for you. Ok, so now you should have a couple perfect squares and one or so imperfect. Good job. Write them out, perfect squares first, biggest to smallest, and then the imperfect. Why? Because Ms. B likes doing that way. Also, you know. Organization. And laziness - it'll be less work for you later. Next, simplify the perfect squares. Easy - that's why they're perfect. Next, multiply the square roots (that you just found) together. Put it aside. Look at the imperfect squares, and multiply those together. Ok, you should only have two numbers now: the square root, and a radical. Cool cool, you're on track. Put them together. Like that's it. You thought this was gonna be hard?? Yeah I mean sorta was because, hey, math. Work. No thanks. But you did it! 1. Factor 2. Find perfect squares 3. Simplify 4. Multiply 5. Simplify Bonus Question: What if you go past a perfect square when you are factoring?? No problem. As we (should) know, perfect square = y X y. So, if you have two of the same factors, you know they can be multiplied to equal a perfect square. Basically, you just skipped a step. No big deal. In fact, good job. Go get yourself a prize. Oh yeah, we also learned about cube roots. But guess what? It's the same thing. Literally the exact same thing. Except all the square roots are cube roots. Perfect squares are perfect cubes. I could copy and paste that whole thing and replace a couple words. Or you could just re read it yourself. Have fun with that. If this was totally confusioning and made no sense, yeah. That's pretty accurate. If you have your math book open next to you while reading through these, it'll probably make more sense... no guarantees. And, watch the video below to understand because this random dude actually explains it in like three minutes... less if you have no patience and speed if up like me oops. And enjoy this video please, all my ads on youtube are for math tutors now. Help. (also this guy is low key creepy. They probably could've chosen a better picture.) Peace out little kiddys.
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alrighty despite falling asleep for a quick sec today i've been put in charge to summarize the lesson so here she is: basically lesson 1.5 was a review of some stuff about exponents that we did last year. do i remember this? lets not talk about it. but today we covered 7 of the 10 rules for exponents. the first one is for exponents of 1 and 0. to summarize, anything with an exponent of 1 is just itself (for example, 5 to the power of 1 is just 5), and anything with an exponent of 0 is simply 1. next is the product rule. this is the rule that says if two of the same number with exponents are being multiplied, all you have to do is keep the original number and change the exponent to the sum of both exponents. for example, if you had 2 to the power of 3 multiplied by 2 to the power of 2, it would be two to the power of 5 (i don't know how to type exponents so we can just stick with writing it out lol). this rule also states that the number that is being multiplied cannot be 0. thiiiiiiird is the quotient rule. this is basically for the reverse of the product rule, where if its division of two of the same number with exponents, you subtract the numerators exponent by the denominators exponent. again, the number cannot be 0. numero quatro is the power rule. i don't know how to use my words and i don't know how to make the little tiny numbers so heres a drawing of an example. its pretty straight forward tho so we chillin ------------------> basically like,,, if a number with an exponent is multiplied by an exponent, just add the exponents together. yikes. rule number 5 is for raising a product to the power. this is when you have an equation in brackets and an exponent outside of the brackets, like (3x)2. this just means 3x • 3x , or (3 • 3) • (x • x). an easy way to solve this is to multiply both things inside the brackets by the exponent separately. again, apparently i can't construct comprehensive sentences so heres a drawing that will hopefully make more sense ↓ also this rule only applies to monomials. ok rule 6: raising a fraction to a power. to do this, take a fraction in brackets with an exponent outside and put the exponent on both the numerator and denominator, and find the final fraction. not even gunna try to write this out so here -------> final rule. thank god. its negative exponents! heres a video to explain these and thats it. what a ride. honestly exhilarating. hope you had fun, i didn't. good luck decoding that horrific page of nonsense. Hey y'all! Yesterday, we learned about squares, square roots, cubes, and cube roots. Here are the basics of all three: To start, here are some symbols and names relating to the unit. √ - This is a radical sign, which represents square roots. A radicand is the number inside the radical sign - √4 (It could be any number.) And the number to the left of the radical sign is the index - ∛ (An index can be any number, and indicates the number of identical factors needed to multiply to make the the number you're looking for.) Also note: any time a radical sign is present without an index, it means it has an index of 2, or more simply that it's a square root. (It takes too long to write out a two every time we have to write a square root.) To square a number is to multiply a number by itself twice. (Eg. 7² = 7 x 7 = 49, With 7² being the squared number.) It's called squaring because if you made a square with a surface area of 49 cm, each side would have 7 cm. A square root is a number that multiplies by itself to create a new number. For example, the square root of 81 is 9, because 9 x 9 = 81. √81 is also a way to symbolize a square root. Square roots don't even have to be rational. The square root of 6 is 2.4494897...etc. because that very long number times itself will still make 6. A perfect square is a number with a rational square root. (Eg. 6 x 6 = 36) Perfect squares can even be fractions, as long as the denominator and numerator separately are perfect squares. (Eg. 25/64 (I'm not sure how to do a fraction on here sorry) is a perfect square. 25 is a perfect square because 5 x 5 = 25 and 8 x 8 = 64.) To determine if a number is a perfect square without a calculator, simply start by finding all the prime factors of that number. (3 x 3 x 5 x 5 = 225) Then, distribute these four factors into two 'teams'. (3 x 5 vs. 3 x 5) If the factors can't be distributed equitably, then it's not a perfect square. Although, if the number is a perfect square, you can find the square root of that number by multiplying the numbers on each team together, and voila, that is the square root of the number. ( 3 x 5 = 15 and 15 x 15 = 225) Next up are cubes and cube roots. These are almost exactly the same as squares and square roots, but with the number 3 instead of 2. To cube a number is to multiply a number by itself three times. (Eg. 2 x 2 x 2 = 8) The cube root of a number is the factor of that number that multiplies itself by 3 to create that number. (The cube root of 8 is 2.) Again, cube roots do not have to be rational. We can identify a cube root by an index of 3 on the radical sign. (∛) Like square roots, there are perfect cubes which are numbers that have rational cube roots. (Eg. 4 x 4 x 4 = 64) Fractions can be perfect cubes as long as the numerator and denominator are perfect cubes individually. To find out if a number is a perfect cube without a calculator, find the prime factors of that number and almost exactly like with square roots, divide the factors into three even 'teams' and multiply them. (Remember: if the factors can't split evenly, it's not a perfect cube.) For example: 5 x 5 x 5 = 125. And there you have it, squares, square roots, cubes, and cube roots! Here's a video to summarize all of the above. (Side note: The way this guy draws his 2s is slightly disturbing.) Today we learned about Greatest Common Factors (GCF) and Least Common Multiples (LCM). A Greatest Common factor is the largest number that can divide into a given set of numbers. For example the Greatest Common Factor for 12, 36 and 80 would be 4. A strategy you can use to find these would be to find the prime factors for the numbers, then list each common factor the least amount of times it appears in one number. (You can find prime factors by making a factor tree where you put the number you want to factor at the top then divide it into numbers that when multiplied make up the original number and repeat for each subsequent number until you end up with only prime numbers.) A least common multiple is the smallest non zero number that two or more whole numbers can be multiplied into. For example the multiple For 8 would be 0, 8, 16, 32, 64. There are three different strategies you may use when trying to find an LCM, These three strategies are found on page 13, 14 and 15. A strategy that I find to work well is the third one, the one Mrs.B prefers ( on page 15). |
Find your green dot!AuthorsWe are members of Esquimalt High School's Gifted Math 10 class. Most of us like Hi-Chews. Archives
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