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

Sep 10, 2017

An example would be:

$2 \sqrt{3} + 5 \sqrt{7} \text{ }$ and $\text{ } 2 \sqrt{3} - 5 \sqrt{7}$

#### Explanation:

I think you just did.

The expressions:

$a \sqrt{b} + c \sqrt{f} \text{ }$ and $\text{ } a \sqrt{b} - c \sqrt{f}$

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:

$\left\{\begin{matrix}a = 2 \\ b = 3 \\ c = 5 \\ f = 7\end{matrix}\right.$

we have:

$2 \sqrt{3} + 5 \sqrt{7} \text{ }$ and $\text{ } 2 \sqrt{3} - 5 \sqrt{7}$

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:

$\left(a \sqrt{b} + c \sqrt{f}\right) \left(a \sqrt{b} - c \sqrt{f}\right) = {\left(a \sqrt{b}\right)}^{2} - {\left(c \sqrt{f}\right)}^{2}$

$\textcolor{w h i t e}{\left(a \sqrt{b} + c \sqrt{f}\right) \left(a \sqrt{b} - c \sqrt{f}\right)} = {a}^{2} b - {c}^{2} f$

and with our choice of coefficients we find:

$\left(2 \sqrt{3} + 5 \sqrt{7}\right) \left(2 \sqrt{3} - 5 \sqrt{7}\right) = {2}^{2} \left(3\right) - {5}^{2} \left(7\right) = 12 - 175 = - 163$