How do you simplify #sqrt(7+sqrt4)#?

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Answer 1 · 2015-07-23T22:13:08

You take the square root of #4# first, then take the square root of the resulting expression.

Notice that your expression contains the square root of #4#, which a perfect square. More specifically,

#4 = 2 * 2 = 2""^2#

This means that you can write

#sqrt(7 + sqrt(4)) = sqrt(7 + sqrt(2""^2)) = sqrt(7 + 2) = sqrt(9)#

This time you come across the square root of #9#, which is another perfect square.

#9 = 3 * 3 = 3""^2#

Therefore, your expression becomes

#sqrt(9) = sqrt(3""^2) = 3#

Put all this together to get

#sqrt(7 + sqrt(4)) = sqrt(7 + 2) = color(green)(3)#