Solve for m in this nice Algebra equation | Math Olympiad Mathematics

Añadido: 2026-06-02

[00:00:01]In this video, let us solve for M given M divided by 2 raised to power 3 is equal to 8.

[00:00:23]We are given M divided by 2 raised to power 3 is equal to 8.

[00:00:33]The first thing we need to do is to separate the elements in this bracket.

[00:00:38]By law of indices, given A over B raised to power P we can open the brackets to give us A raised to power P divided by B raised to power P.

[00:00:54]This would then give us M raised to power 3 divided by 2 raised to power 3.

[00:01:02]So, we have M raised to power 3 over 2 raised to power 3 is equal to 8.

[00:01:13]Next, let us go ahead and cross-multiply so that we have M raised to power 3 times 1 is equal to 2 raised to power 3 times 8 here we can also write as 2 raised to power 3.

[00:01:34]So, this gives us M raised to power 3 is equal to 2 raised to power 3 squared.

[00:01:46]By law of indices A raised to power P times Q is the same thing as A raised to power Q times P which is also the same thing as A raised to power Q raised to power P.

[00:02:04]In other words, we can switch these powers in the bracket to give us M raised to power 3 equal to 2 raised to power 2.

[00:02:17]Then raised to power 3. We now have M raised to power 3 equal to 2 raised to power 2 is 4 then raised to power 3.

[00:02:30]I'm going to move this to the left, so we have M raised to power 3 - 4 raised to power 3 is equal to 0.

[00:02:42]This is difference of two cubes.

[00:02:44]By identity given A raised to power 3 - B raised to power 3 we can factorize this to give us A - B into bracket A squared + AB + B squared.

[00:03:05]Now, using this to factorize this will give us M - 4 into bracket M squared + 4 M + 4 raised to power 2 equals to 0.

[00:03:27]We can tidy this up to give us M - 4 into bracket M squared + 4 M + this will be 16 equals to 0.

[00:03:44]Which will then imply that either M - 4 is equal to 0 or M squared + 4 plus 16 is equal to zero.

[00:04:02]From here, m is equal to four.

[00:04:05]We'll call this m1.

[00:04:08]And then we proceed to solve this quadratic equation for two other values of m.

[00:04:15]If we compare this with ax squared plus bx plus c, then we see a is equal to one, b is equal to four, and c is equal to 16.

[00:04:32]Then applying the quadratic formula, we're going to have m is equal to minus b plus or minus b squared minus 4ac then divided by two times a.

[00:04:51]Already have values for a, b, and c.

[00:04:54]Therefore, m is equal to minus b. B is four.

[00:05:00]Then plus or minus square root of four squared minus four times a. A is one.

[00:05:10]And c is 16.

[00:05:15]Then divided by two times a. A is one.

[00:05:22]This will give us m is equal to negative four plus or minus square root of this will be 16 minus four times one is four.

[00:05:36]So, we can say four times 16.

[00:05:40]Square root of that, then divided by two times one is two.

[00:05:45]Giving us m is equal to -4 plus or minus square root of 16 is common here.

[00:05:55]So, we have 16 into 1 - 4 then all divided by 2.

[00:06:05]m is equal to -4 plus or minus square root of 16 * -3 divided by 2.

[00:06:21]We can go ahead and separate this radical to give us m equal to -4 plus or minus square root of 16 * square root of -3 then then divided by 2.

[00:06:41]Given us m is equal to -4 plus or minus square root of 16 is 4 square root of -3 is square root of -1 * 3 and square root of -1 is i, so we have 4i root 3 here then divided by 2 to give us m equal to -4 over 2 plus or minus 4i root 3 divided by 2.

[00:07:20]2 here 1, 2 here 2 2 here 1, 2 here 2.

[00:07:26]So, we have m is equal to -2 plus or minus 2i root 3.

[00:07:38]So, here we have we already got value for M1 which gave us 4.

[00:07:47]Now, if we split this into, we get M2 is equal to -2 + 2i root 3 and then M3 is equal to -2 - 2i root 3.

[00:08:10]Giving us our three possible solutions to this problem.

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[00:08:26]Bye.