# How do you write two binomial in the form #asqrtb+csqrtf# and #asqrtb-csqrtf#?

##### 1 Answer

#### Answer:

An example would be:

#2sqrt(3)+5sqrt(7)" "# and#" "2sqrt(3)-5sqrt(7)#

#### Explanation:

I think you just did.

The expressions:

#asqrt(b)+csqrt(f)" "# and#" "asqrt(b)-csqrt(f)#

are already binomials, so it would seem that the answer is in the question.

I am not sure what is really wanted, except that we could substitute numerical values for the variables.

For example, with:

#{(a=2),(b=3),(c=5),(f=7):}#

we have:

#2sqrt(3)+5sqrt(7)" "# and#" "2sqrt(3)-5sqrt(7)#

What is interesting about these expressions is that they are radical conjugates of one another. If you mutiply the two binomials together you will get a rational result (assuming the coefficients are rational).

In general, we find:

#(asqrt(b)+csqrt(f))(asqrt(b)-csqrt(f)) = (asqrt(b))^2-(csqrt(f))^2#

#color(white)((asqrt(b)+csqrt(f))(asqrt(b)-csqrt(f))) = a^2b-c^2f#

and with our choice of coefficients we find:

#(2sqrt(3)+5sqrt(7))(2sqrt(3)-5sqrt(7)) = 2^2(3)-5^2(7) = 12-175 = -163#