How do you simplify #(3+sqrt7)(3-sqrt7)#?

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Answer 1 · 2015-09-06T23:16:36

#(3 + sqrt(7))(3 - sqrt(7)) = 2#

You can simplify this expression by using the formula for the difference of squares

#color(blue)(a^2 - b^2 = (a-b)(a+b))#

In your case, you ca say that #a = 3# and #b = sqrt(7)#. This means that you have

#(3 + sqrt(7))(3-sqrt(7)) = 3^2 - (sqrt(7))^2 = 9 - 7 = color(green)(2)#

Alternatively, if you don't know the formula for the difference of squares, you can rewrite this expression by expanding the parantheses

#(3 + sqrt(7))(3-sqrt(7)) = 3 * 3 -color(red)(cancel(color(black)(3 * sqrt(7)))) + color(red)(cancel(color(black)(3 * sqrt(7)))) - sqrt(7) * sqrt(7)#

Once again, the result will be

#3 * 3 - sqrt(7) * sqrt(7) = 3^2 - 7 = color(green)(2)#