Perform the indicated operations and/or simplify each expressi...

Question

Perform the indicated operations and/or simplify each expression. Assume all variables represent positive real numbers. $\frac{-4}{\sqrt[3]{3}} + \frac{1}{\sqrt[3]{24}} - \frac{2}{\sqrt[3]{81}}$

Accepted Answer

Hello, today we're gonna be simplifying the given expression. So what we are given is negative six over the cube root of four plus four over the cube root of 108 minus seven over the cube root of 256. So before we start to simplify this, let's go ahead and simplify each of the given denominators While the Cube root of four is in its simplest form. So we're gonna go ahead and leave that alone In the second term. We have the cube root of 108 and 108 is not a perfect cube. So we can go ahead and rewrite this as a product of prime numbers. Now, 100 and eight can be rewritten as four times 27. And I know we said we want prime numbers but notice how we have the cube root of four in the first term. So we're going to want to have a four in the denominator of the cube root in every other term. So we're gonna circle this four and leave it alone for now. Now 27 can be written as nine times three where three is a prime number and nine can be rewritten as three times three, which are both prime numbers. So that means we can go ahead and rewrite the cube root of 100 and eight as the cube root of four times three times itself three times. So that's gonna be four times three. Cute. Now, the cube root allows us to take out any numbers that are multiples of three, which in this case here three times itself three times is a multiple three. So we can go ahead and take out three cubed as a single three in front of the cube root. And that's going to give us three times the cube root of four, which is the simplified denominator of the second term. Now, in the third term, 256 can be rewritten or simplified using a number tree And 256 can be rewritten as four times 64. And just like before, we want to have a four in the cube root. So we're gonna leave this for alone for now, 64 can be rewritten as eight times and eight can be written as four times to where two is a prime number And four can be written as two times 2, which are both prime numbers. So the cube root of 256 can be rewritten as the cube root of four times two cubed times two cubed. And again, we can go ahead and take out any numbers that are multiplied by itself three times, which in this case, we have two instances of two cubed. So we can take out both two cubes as two times two times the cube root of four and two times two is going to give us four. So we have four times the cube root of four, which is going to be the simplified form of the denominator of the third term. Now let's go ahead and put this back into the original expression. And that's going to give us negative six over the cube root of four plus four Over three times the cube root of four minus 7/4 times the cube root of four. Now, what we want to go ahead and do is combine the three fractions together. And in order to do that, we're going to need to find the lowest common denominator because we can only subtract and add fractions together if they have the same denominator. Now notice that each of the denominators has the cube root of four. So the lowest common denominator is going to need to have a cubit of four in it. But let's take a look at the coefficients of the denominators. In the first term, we have the coefficient of one, the second term, we have the coefficient of three. And then the third term we have the coefficient of four. So what's gonna be the lowest common denominator or the lowest common multiple of 13 and four. Well, that's going to be 12. So we're going to multiply the first term by 12/12. So that way we can get 12 as the coefficient in the denominator, We're going to multiply the second term by 4/4. And finally, we're going to multiply the last term by 3/3. So multiplying this together 12/12 times, the first term is going to give us negative 72/ times. The key root of four, multiplying the second term by 4/4 is going to give us 16/12 times the cube root of four. And in the last term, multiplying it by 3/3 is going to give us minus 21/12 times the cube root of three. And now that we have the same denominator in each of the fractions, we can go ahead and combine it onto a single fraction. So what we need to do now is go ahead and combine the numerator negative 72 plus 16 minus 21 is going to give us negative 77 all over 12 times the cube root of four. And now we want to go ahead and rationalize the denominator because we don't want to have a cube root in the denominator. In order to rationalize this, let's go ahead and multiply the numerator and denominator by the cube root of four squared. Now, the reason why we want to multiply by the cube root of four squared is because in the denominator, if we multiply the cube root of four by the cube root of four squared, we can go ahead and combine these together to give us the cube root of four times four squared, which is going to give us four cubed. And just like before, we can go ahead and take out four cubed as a single four because of the cube root. So the cube root of four cubed is going to simplify to give us four. So if we multiply the numerator is together, what we have is negative 77 times the key root of four squared. And in the denominator, we have 12 cube root of four times the cube root of four squared, which is going to simplify to give us 12 times four. Now, finally, we're just going to simplify the number in the numerator four squared is going to give us 16. So we have negative 77 times the cube root of 16. And in the denominator 12 times four is going to give us 48. So the simplified version of the expression is going to be negative times the cube root of 16 all over 48. And with that being said, the answer to this problem is going to be C 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.

Perform the indicated operations and/or simplify each expression. Assume all variables represent positive real numbers. $\frac{-4}{\sqrt[3]{3}} + \frac{1}{\sqrt[3]{24}} - \frac{2}{\sqrt[3]{81}}$

Key Concepts

Cube Roots and Radicals

Simplifying Radicals

Adding and Subtracting Radical Expressions

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Key Concepts

Cube Roots and Radicals

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Perform the indicated operations and/or simplify each expression. Assume all variables represent positive real numbers. $\frac{-4}{\sqrt[3]{3}} + \frac{1}{\sqrt[3]{24}} - \frac{2}{\sqrt[3]{81}}$