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

Question

Abigail Allen · Accepted Answer

An example would be:

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

I believe you just did. The terms #asqrt(b)+csqrt(f)" "# and #" "asqrt(b)-csqrt(f)# are already binomials, indicating that the question itself appears to contain the answer. Apart from the fact that we could replace the variables with numerical values, I'm not sure what is actually desired. As an illustration, with: #{(a=2),(b=3),(c=5),(f=7):}# we have: #2sqrt(3)+5sqrt(7)" "# and #" "2sqrt(3)-5sqrt(7)# These expressions are intriguing because they are radical conjugates of each other; multiplying the two binomials together will yield a rational result, provided the coefficients are rational. Generally speaking, we discover: #(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 using the coefficients we have selected, we find: #(2sqrt(3)+5sqrt(7))(2sqrt(3)-5sqrt(7)) = 2^2(3)-5^2(7) = 12-175 = -163#

Jace Anderson · Answer

undefined To write two binomials in the form asqrtb+csqrtf and asqrtb-csqrtf, you can follow these steps:

1. Identify the values of a, b, c, and f in the given binomials.
2. Simplify the square roots of b and f, if possible.
3. Write the binomials in the desired form by separating the square roots and coefficients.

For example, let's say we have the binomials (3√2 + 2√5) and (3√2 - 2√5):

1. In this case, a = 3, b = 2, c = 2, and f = 5.
2. The square root of 2 is already simplified, but the square root of 5 cannot be simplified further.
3. Writing the binomials in the desired form, we have:
   - (3√2 + 2√5) = 3√2 + 2√5
   - (3√2 - 2√5) = 3√2 - 2√5

So, the binomials in the form asqrtb+csqrtf and asqrtb-csqrtf are 3√2 + 2√5 and 3√2 - 2√5, respectively.