Concept Check: By what number should the numerator and denominato...

Question

Concept Check: By what number should the numerator and denominator of 1/(∛3 - ∛5) be multiplied in order to rationalize the denominator? Write this fraction with a rationalized denominator.

Accepted Answer

Hello, today we're going to be rationalizing the denominator of the given expression. So what we are given is one over the cube root of 11 minus the cube root of seven. Now the easiest way to approach this problem is to utilize the difference of cubes. The difference of cube states that if you have the quantity A cubed minus B cubed, this is going to factor to a minus B times the quantity A squared plus A times B plus B squared. Now, in order to utilize this, let's go ahead and allow a two equal to the cube root of 11. And we're going to allow be to equal to the cube root of seven. So if we plug this in to the difference of cubes, what we end up with is the cube root of 11 cubed which is going to cancel that just give us 11 minus the cube root of seven cubed, which is just going to leave us with seven. And that is going to equal to the cube root of 11 minus the cube root of seven times the quantity the cube root of squared, which is going to give us the cube root of 11 squared plus the cube root of 11 times the cube root of seven, which is going to give us the cube root of 11 times seven plus the quantity the cube root of seven squared, which is going to give us the cube root of seven squared. Now, just to simplify this statement, on the left hand side, 11 -7 is going to give us four. And on the right hand side, 11 times seven is going to give us 77. So again, we're going to be left with four is equal to the cube root of 11 minus the cube root of seven times the cube root of 11 squared plus the cube root of 77 plus the cube root of seven squared. Now you may be thinking, how are we going to utilize this? This seems very complicated while there's one thing to notice in the given statement, we have the cube root of 11 minus the cube root of seven in the denominator. And according to this rule, we have the cube root of 11 minus the cube root of seven times this entire quantity and that is going to equal to four. So if we wanted to rationalize the original expression, we can go ahead and multiply the numerator and denominator by this try no meal here that will allow us to simplify the denominator to just four. So let's go ahead and do that. So the original statement is one over the Q Bert of 11 minus the Q bert of seven. And since we're gonna be rationalizing this, we want to go ahead and multiply the numerator and denominator by the cube root of 11 squared plus the cube root of 77 plus the cube root of seven squared. And again, we're going to multiply the denominator by the same thing, The keyboard of 11 squared plus the keyboard of plus the cube root of seven squared. So when we multiply these values together in the denominator, this argument Matches the argument of the difference of Cubes. So we can go ahead and replace the denominator with four. And this is going to simplify to give us the cube root of 11 squared plus the cube root of 77 plus the cube root of seven squared. And that is going to be over four. And finally, we just want to go ahead and simplify the numbers in the numerator 11 squared is going to simplify to give us 121. So we have the cube root of 121 plus the cube root of 77 Plus the cube root of seven squared, which is going to give us 49. And that's all going to be over four, which is going to be the simplified form of the given expression. And with that being said, the answer to this problem is going to be D So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

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Concept Check: By what number should the numerator and denominator of 1/(∛3 - ∛5) be multiplied in order to rationalize the denominator? Write this fraction with a rationalized denominator.