Factor out the least power of the variable or variable express...

Question

Factor out the least power of the variable or variable expression. Assume all variables represent positive real numbers. See Example 8.

$(3 r + 1)^{- \frac{2}{3}} + (3 r + 1)^{\frac{1}{3}} + (3 r + 1)^{\frac{4}{3}}$

Accepted Answer

Welcome back. I am so glad you're here. We are given this expression, parentheses five Q plus two raised to the negative 2/5 plus parentheses five Q plus two raised to the 3/5 plus parentheses five Q plus two raised to the 8/5 and essentially we are asked to factor this, we're going to be isolating the common factor with the lowest power in factoring that out. So what does that mean? The common factor with the lowest power? Well all three of these have a five Q plus two but they have different powers. The 1st 15 Q plus two raised to the negative 2/5. The second one, five Q plus two. This one is raised to positive 3/5 and the 3rd +15 Q plus two is raised to the positive 8/5. So the lowest power when we compare all three of those exponents, those degrees. Those power is the negative 2/5. So what we do is we're factoring out of five Q plus two, raised to the negative 2/5. Or in other words were dividing all three of these terms by five Q plus two, raised to the negative 2/5. And so now we just have to do that. Work of dividing them to see what is left. After factoring out the five Q plus two, raised to the negative 2/5 when we have terms that are raised to exponents and they're being divided. We are going to as we recall from previous lessons, we are going to subtract the exponents. So for this first one we're taking negative 2/5 minus negative 2/5. Or in other words this is negative 2/5 plus 2/5. We have a common denominator so we can add the numerator is negative, two plus two is zero. So 0/5 switch equals zero. Or in other words we've got five Q plus two raised to the zero power. Well anything raised to the zero is equal to one. So we've got a one for our first term after we factored out the five Q plus two raised to the negative 2/5. Let's factor five Q plus two, raised to the negative 2/5 out of five Q plus two raised to the 3/5. So again we're going to subtract the exponents 3/ minus negative 2/5. Or in other words 3/5 plus 2/5 common denominators. So we can add our numerator is three plus two is five. So we've got 5/5 which equals one. Or in other words we've got five Q plus two raised to the first which is five Q plus two and we can have plus five Q plus two down here. Okay, we got the first two terms factored let's factor the third term again. We're going to subtract our exponents 8/5 minus negative 2/5 or in other words 8/5 plus 2/5 common denominators. So we can add the numerator eight plus two is 10. We've got 10 15 which equals two. Or in other words this is five Q plus two. With the new exponent of two or five Q plus two squared. So we can put down here plus five Q plus two squared. All right now we have factored out that common factor with the lowest power. Now we just have to simplify what's in the brackets there. So I'll bring down this five Q plus two, raised to the negative 2/5 and let's do the work in the brackets. Bring down that one plus and we can get rid of the parentheses around this five Q plus two and four or five Q plus two squared. We need to foil that out. So that will be five Q plus two times five Q plus two. And bringing all of this down five Q plus two raised to the negative 2/5 times one plus five Q plus two plus. Now let's spoil this first five q times 5 to 5 Q. Is going to be a 25 Q squared. Outsides five Q times two is plus 10-Q insides, two times five Q. Is plus 10-Q. And last two times two is plus four. Now we can combine our like terms will bring down five Q plus two, raised to the negative 2/5 again. And let's start with the degree with the highest term or the term with the highest degree which is this 25 Q squared. And the next highest degree is just our Q. So we have five Q plus 10-Q plus 10-Q. That is plus 25 Q. And now our constant terms one plus two plus four is plus seven. And that's it. We have simplified it. Now we're looking for an answer choice that is five Q plus two raised to the negative 2/ times 25 Q squared plus 25 Q plus seven. That matches with answer choice C. Well done. We'll catch you on the next one.

Factor out the least power of the variable or variable expression. Assume all variables represent positive real numbers. See Example 8.
$(3 r + 1)^{- \frac{2}{3}} + (3 r + 1)^{\frac{1}{3}} + (3 r + 1)^{\frac{4}{3}}$

Key Concepts

Exponent Rules

Factoring Expressions with Variables

Properties of Positive Real Numbers

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