150-300-Word Response to the Following Discussion Question

150-300-word response to the following discussion question

In your own words, explain the first condition that must be met for a simplified radical. Explain why 5/sqrt(2) is not simplified and demonstrate the steps we must take to simplify it.

For a radical expression to be simplified, all the roots must be in the numerator (the top) of the fraction, if there is a fraction. There cannot be any roots in the denominator (the bottom) of the fraction. If there are roots in the denominator, you need to rationalize it. In addition, if you have something like sqrt(20), you can simplify that by factoring the 20: sqrt(4*5), splitting the root: sqrt(4)*sqrt(5), and evaluating the first root: 2*sqrt(5). The example 5/sqrt(2) is not simplified, since the square root of 2 is in the denominator. The basic method for simplifying things like this is to multiply the top and bottom of the expression by something to cancel out the roots on the bottom. In this case, we use sqrt(2)/sqrt(2).

5/sqrt(2) * sqrt(2)/sqrt(2)

= 5*sqrt(2) / sqrt(2)*sqrt(2)

Now the square roots of 2 on the bottom can be combined into:

= 5*sqrt(2) / 2

It is now rationalized, since there are no roots in the denominator.

We can’t simplify it any further, since there is no way to simplify the square root of 2.

question two

150-300-word response to the following discussion.

In your own words, what are radical expressions? What is the process we follow when adding, subtracting, multiplying, and dividing rational expressions? In your answer, demonstrate the process for each one with your own example.

A radical expression is an expression that has square roots, cube roots, and other higher order roots. These expressions could have variables, powers, and numbers in them. To add or subtract radical expressions, we need to make sure we have “like roots”, which is similar to when we added and subtracted polynomials and needed “like powers”. While at that point, we were looking at the power, now we look at the order of the root. For example, you cannot add or subtract cuberoot(2) and sqrt(2), since the order of the roots are different. However, you can add sqrt(2) and sqrt(2) to get 2sqrt(2), since the roots are like.

Addition Example:

sqrt(108) + sqrt(12)
= sqrt(3*36) + sqrt(3*4)
= 6sqrt3 + 2sqrt3
= 8 sqrt 3

Subtraction:

sqrt(108) - sqrt(12)
= sqrt(3*36) - sqrt(3*4)
= 6sqrt3 - 2sqrt3
= 4 sqrt 3

To multiply or divide roots, you don’t have to have like roots, just like with polynomials, you didn’t have to have like powers. You multiply or divide the values inside the roots separately from the values outside the roots.

Division Example:

sqrt(8) / sqrt(2)

= sqrt(8/2)

= sqrt(4)

= 2

Multiplication:

sqrt(8) * sqrt(2)

= sqrt(8*2)

= sqrt(16)

= 4