In Exercises 93–102, factor and simplify each algebraic expressio...

Question

In Exercises 93–102, factor and simplify each algebraic expression. (x+5)^−1/2−(x+5)^−3/2

Accepted Answer

Hey everyone here we are asked to factor and simplify the given expression of the quantity of X plus one to the negative 1/5 power minus the quantity of X plus one to the negative 6/5 power. Now the first step here is to determine our greatest common factor and we want to note that both terms have the same base. So our greatest common factor will be the term with the smaller exponents. And since both exponents are negatives, our 6/5 is less than our 1/5. Therefore the greatest common factor is the quantity of X plus one to the negative 6/5 power. Now the next step is to determine what the greatest common factor has to be multiplied by so that our factored expression is equivalent to our original expression. So taking a look at our first term here of the quantity of X plus one to the negative 1/5 power. We can rewrite this as the quantity of X plus one to the power of negative 6/5 plus 5/ here we can use the product rule for exponents to rewrite this expression as the quantity of X plus one to the negative 6/ power times the quantity of X plus one to the 5/5 power. So this expression tells us that the greatest common factor once multiplied by the quantity of X plus one to the 5/5 power will return us our initial term in our original expression. So we can now include this quantity of X plus one to the 5/5 power in our factored expression. And now we can move on to our second term and note that the only difference between our greatest common factor and this second term is this negative sign here. So all we have to do is include this negative one within the parentheses of our factored expression because our greatest common factor times negative one returns our second term. So from here we must note that this 5/5 exponents is equivalent to one. So our factored expression here can be rewritten as the quantity of X plus one to the negative 6/5 power times the quantity of X plus one minus one here, our ones cancel. So our simplified expression is now the quantity of X plus one to the negative 6/5 power times X. Now using the negative exponents rule, we can rewrite this expression as the X divided by the quantity of X plus one to the positive 6/ power, which is our answer here for choice. B Thanks for watching. I hope you found this video helpful and I'll see you again next time.

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