Simplifying Cube Roots (2 Ways) | Perfect Cube Factors and Prime Factorization | Math with Mr. J

welcome to math with Mr J [Music] this video I'm going to cover how to simplify cube roots two different ways both ways are very similar and do the exact same thing but they are slightly different approaches we're going to take a look at the same cube roots and simplify them using both methods we're going to start by looking for factors that are perfect cubes and then take a look at using prime factorization let's jump into number one where we have the cube root of 48. now 48 is not a perfect Cube so we're not going to get a nice clean cut whole number for our answer but we can simplify this cube root we can break it down so to speak we're going to do this by looking for factors of 48 that are perfect cubes and then find their cube root we will then pull that cube root out now to the right I have the first 10 perfect cubes knowing and being familiar with these or as many of these as you can will make simplifying cube roots and working with cube roots in general much easier so are any of the first 10 perfect cubes a factor of 48 yes 8 is a factor of 48. we know that 8 times 6 equals 48 8 and 6 are factors of 48. they go into 48 so to speak so let's rewrite the cube root of 48 as the cube root of eight times six now the multiplication or product property of cube root lets us split this into the cube root of eight times the cube root of six meaning we have the cube root of eight times the cube root of 6. this is still equivalent to the cube root of 48. we're not changing the value of anything at all now we can take the cube root of eight which is two two times two times two equals eight so again the cube root of eight is two so we end up with two times the cube root of 6 and that's our final simplified answer we can't simplify or break this down any further so again our final simplified answer 2 times the cube root of 6. let's move on to number two where we have the cube root of 250. 250 is not a perfect Cube but we can simplify we need to look for factors of 250 that are perfect cubes well 125 times 2 equals 250 and 125 is a perfect Cube since there's a factor of 250 that's a perfect Cube we can simplify we can rewrite this as the cube root of 125 times 2. now let's split this into the cube root of 125 times the cube root of 2. now we can take the cube root of 125 which is 5 5 times 5 times 5 equals 125. again the cube root of 125 is 5. so this is going to equal five times the cube root of 2 and that's our final simplified answer 5 times the cube root of 2. we cannot simplify or break that down any further so there's how to simplify a cube root by looking for factors that are perfect cubes let's move on to simplifying cube roots using prime factorization here are our examples and these are the same exact ones we just did but let's take a look at a different approach let's jump into number one where we have the cube root of 48. we're going to do this by using the prime factorization of 48 to pull out any cube roots of perfect cubes so let's start with the prime factorization of 48. we need to break this down until we only have prime factors now there are different paths to take in order to get to the prime factors but in the end no matter what path you take you'll get to the same prime factors so something to keep in mind let's come to the side here and take a look at the prime factorization of 48. I'm going to start with the factors of 2 and 24. now 2 is prime so we are done over there and we can continue to break 24 down let's use the factors of 2 and 12. 2 is prime so we are done there but we can break 12 down let's use the factors of 2. and 6. 2 is prime so we are done there but we can break 6 down let's use the factors of 2 and three two and three are both Prime so we are done that's the prime factorization of 48. now that we have the prime factorization of 48 we can rewrite the cube root of 48 with these factors we can rewrite this as the cube root of 2 times two times two times two times three this is still equivalent to 48 so we're not changing the value of anything now we need to find prime factors in groups of three we have a group of three twos right here that represents a perfect Cube that we can pull out two times two times two equals eight and the cube root of eight is two so let's pull out a two so this is going to equal 2 times the cube root of two times three we still have that 2 times 3 under the cube root symbol we cannot simplify any further so let's multiply we have two times three which equals six so we end up with two times the cube root of 6 and that's our final simplified answer 2 times the cube root of six let's move on to number two where we have the cube root of 250. 250 is not a perfect Cube so let's simplify this cube root we will start with the prime factorization of 250. let's come to the side and take a look at the prime factorization of 250. I'm going to start with the factors of 2 and 125. 2 is prime so we are done there 125 we can break that down let's use the factors of 5 and 25 5 is prime so this is done but 25 we can break down let's use the factors of 5 and 5. 5 is prime so we are done that's the prime factorization of 250. so we can rewrite the cube root of 250 S2 times 5 times 5 times 5. now we need to look for any prime factors in a group of three well we have three fives right here so we have a group of three fives that represents a perfect Cube that we can pull out five times five times five equals one hundred twenty-five and the cube root of 125 is five we can pull out a five so we now have five times the cube root of two we cannot break that down any further we cannot simplify any further so this is our final simplified answer 5 times the cube root of two so there you have it there are two different approaches to simplifying cube roots I hope that helped thanks so much for watching until next time peace