Question

Express the product left parenthesis, square root of, 10, end square root, minus, 3, right parenthesis, left parenthesis, square root of, 10, end square root, minus, 3, right parenthesis(
10
​
−3)(
10
​
−3) in simplest form.

Answer 1

To express the product \( (\sqrt{10} - 3)(\sqrt{10} - 3) \) in simplest form, we can recognize that this is the square of a binomial. Specifically, it can be rewritten as:

\[
(\sqrt{10} - 3)^2
\]

We can expand this using the formula for the square of a binomial, \( (a - b)^2 = a^2 - 2ab + b^2 \), where \( a = \sqrt{10} \) and \( b = 3 \):

\[
(\sqrt{10})^2 - 2(\sqrt{10})(3) + 3^2
\]

Calculating each term:

1. \( (\sqrt{10})^2 = 10 \)
2. \( 2(\sqrt{10})(3) = 6\sqrt{10} \)
3. \( 3^2 = 9 \)

Putting it all together, we have:

\[
10 - 6\sqrt{10} + 9
\]

Now, simplify:

\[
10 + 9 - 6\sqrt{10} = 19 - 6\sqrt{10}
\]

Thus, the product \( (\sqrt{10} - 3)(\sqrt{10} - 3) \) in its simplest form is:

\[
\boxed{19 - 6\sqrt{10}}
\]