Question

express the product \\((\sqrt{3} + \sqrt{7})(\sqrt{3} + \sqrt{7})\\) in simplest form.

Explanation:

Step1: Recognize the form

The expression \((\sqrt{3}+\sqrt{7})(\sqrt{3}+\sqrt{7})\) is a square of a binomial, which can be written as \((\sqrt{3}+\sqrt{7})^2\).

Step2: Apply the formula \((a + b)^2=a^2+2ab + b^2\)

Here, \(a = \sqrt{3}\) and \(b=\sqrt{7}\). So we have:
\(a^2=(\sqrt{3})^2 = 3\)
\(2ab = 2\times\sqrt{3}\times\sqrt{7}=2\sqrt{21}\)
\(b^2=(\sqrt{7})^2 = 7\)

Step3: Sum the terms

Now, add \(a^2\), \(2ab\) and \(b^2\) together:
\(3 + 2\sqrt{21}+7\)
Combine the like terms (the constant terms):
\(3 + 7+2\sqrt{21}=10 + 2\sqrt{21}\)

Answer:

\(10 + 2\sqrt{21}\)