Simplify. See Example 9. (1/2)/(1 - (√5/2))

Question

Accepted Answer

Welcome back. Everyone in this problem. We want to simplify the expression of a quarter divided by one minus a quarter of the square root of 17. For our answer choices, A is negative four plus the skirt of 17 B is negative four minus the skirt of 17 C is four minus the skirt of 17 and D is four plus the square root of 17. Now, what do we know here? Well, we know that we want to simplify our expression, but it will be much easier to simplify if we have our expression written as one caution instead of a caution being divided by the other. So let's go ahead and do that. So we have here 1/ divided by one minus the skirt of 17, divided by four. Now, what does this really mean? Let's try to write this question as one expression. So we're dealing with a fraction here and one minus the skirt of 17 divided by four. If we think about both of these, they have a common denominator of four and one multiplied by four is four. So I can rewrite this as four and four into four goes one time. The square of 17, multiplied by one is a square of 17. So as a single fraction, it would have been four minus the square of 17, all divided by four. So I can rewrite it as 1/4 divided by four minus the script of 17, divided by four. And using what I know about dividing fractions, I can rewrite it further as 1/4 multiplied by four, divided by four minus the square root of 17. Both terms have a common factor of four. And when we factor it out, we are left with one divided by four minus the square of 17. This looks much simpler than the expression we started with, but we still haven't simplified it completely because we have a radical nr denominator to get rid of that radical, we can rationalize the denominator. And what that means is we need to multiply our denominator by something that will remove the radical. The best way to do that is to multiply our denominator or which is at the difference of a radical by its conjugate require that the conjugate of an expression in that form in the form A minus the square of B would have been equal to a, the same integral part plus the square of B, the opposite radical part. So in this case, that tells us the conjugate of four minus the square of 17 would be four plus the square root of 17. So I can multiply my expression one divided by four minus the square of by four minus the spirit of 17 in the numerator and the denominator or four plus the spirit of 17. Sorry in both, because that would help to keep the same value. Now, why do we do it? Well, no notice that the denominators are the difference of two squares because we are multiplying as some by the difference of the same values in the expression. So we can rewrite it as a difference of two squares. And that's good because now we can find the square of the square of which removes the square root sign, thus removing the radical. Now one multiplied by four plus the square of 17 is four plus the square of 17. And in our denominator, we'd have four squared minus the square root of 17 squared, four squared is 16 and the square root of 17 squared is 17, still not finished yet because 16 minus equals negative one. OK. And no, we can divide both terms by negative 14, divided by a negative one is negative four seven. The square root of 17 divided by a negative one is a negative of a square root of 17. So our answer is negative four minus the square root of 17. And when we look back in our answer choices that would have been answer choice. B thanks a lot for watching everyone. I hope this video helped

Simplify. See Example 9. (1/2)/(1 - (√5/2))

Key Concepts

Rationalizing the Denominator

Conjugates of Binomials

Simplifying Radical Expressions

Watch next