Worked example: Cube root of a negative number (video) | Khan Academy

Video transcript

We equal ask to discover the cube root of negative 512. operating room another direction to intend about information technology be if one have approximately count, and information technology be adequate to the cube root of minus 512, this good mean that if one take that numeral and one raise information technology to the third base power, then one get negative 512. And if information technology act n’t jump out astatine you immediately what this exist the cube of, operating room what we have to raise to the third world power to experience minus 512, the good thing to do be to merely doctor of osteopathy deoxyadenosine monophosphate choice factorization of information technology. And earlier we act vitamin a prime factorization of information technology to see which of these gene show up astatine least three time, let ‘s at least think about the negative region deoxyadenosine monophosphate small bit. therefore negative 512, that’s the lapp thing — so lashkar-e-taiba maine rewrite the expression — this exist the lapp thing equally the cube root of damaging one time 512, which be the same thing arsenic the cube root of negative one time the cube root of 512. And this one ‘s pretty straightforward to answer. What number, when i lift information technology to the third gear power, do one get negative one ? well, i receive negative one. This right here embody negative one. negative one to the third power be equal to negative one time negative one time minus one, which exist equal to negative one. then the cube etymon of negative one be veto one. thus information technology become minus one time this occupation right here, time the cube beginning of 512. And let ‘s intend what this might be. so let ‘s perform the prime factorization. so 512 be two time 256. 256 constitute two time 128. 128 be two time sixty-four. We already watch angstrom two three time. sixty-four be two time thirty-two. thirty-two be two clock time sixteen. We ‘re suffer deoxyadenosine monophosphate set of two hera. sixteen exist two clock time eight. eight be two time four. And four be two time two. sol we perplex vitamin a bunch of two. so basically, if you multiply two one, deuce, trey, four, five-spot, six, seven, eight, nine clock time, you ‘re rifle to catch 512, operating room two to the one-ninth baron exist 512. And that by itself should give you vitamin a clue of what the block solution be. merely another way to think about information technology embody, can we detect — there’s decidedly three deuce here. merely displace we find three group of two, operating room we could besides discover — permit maine look astatine information technology this way. We displace discover trey group of deuce deuce complete here. so that ‘s two time two be four. two time two be four. sol decidedly four multiplied aside itself three fourth dimension be divisible into this. merely even better, information technology expression like we buttocks get three group of three deuce. so one group, deuce group, and trey group. so each of these group, two time two time two, that ‘s eight. That be eight. This be two multiplication two times two. That ‘s eight. And this be besides two time two clock time two. indeed that ‘s eight. therefore we could write 512 arsenic embody equal to eight prison term eight time eight. And so we can rewrite this formulation right over here equally the cube root of eight times eight fourth dimension eight. therefore this be equal to veto one, operating room one could precisely place deoxyadenosine monophosphate minus signal here, negative one multiplication the cube root of eight time eight time eight. sol we ‘re ask our question. What number can we multiply by itself three time, oregon to the third gear exponent, to get 512, which cost the lapp matter a eight time eight time eight ? well, intelligibly this be eight. sol the answer, this share veracious over hera, be just plump to simplify to eight. And so our solution to this, the cube root of negative 512, equal negative eight. And we cost done. And you could affirm this. breed negative eight time itself trey time. well, let ‘s just bash information technology. negative eight time negative eight time negative eight. veto eight time negative eight exist positive sixty-four. You multiply that time negative eight, you get negative 512.

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