How do you simplify #4+sqrt200 + 8sqrt49 - 2sqrt2#?

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Answer 1 · 2015-10-09T00:13:01

#4 * (15 + 2sqrt(2))#

Your starting expression looks like this

#4 + sqrt(200) + 8sqrt(49) - 2sqrt(2)#

The first thing to notice here is that #49# is actually a perfect square

#49 = 7 * 7 = 7^2#

This means that the expression can be first simplified to

#4 + sqrt(200) + 8 * sqrt(7^2) - 2sqrt(2)#

#4 + sqrt(200) + 8 * 7 - 2sqrt(2)#

#4 + sqrt(200) + 56 - 2sqrt(2)#

#60 + sqrt(200) - 2sqrt(2)#

Now take a look at #sqrt(200)#. You know that #200# is not a perfect square, but that you ccan write it as

#200 = 100 * 2 = 10^2 * 2#

This means that you have

#60 + sqrt(10^2 * 2) - 2sqrt(2)#

#60 + 10sqrt(2) - 2sqrt(2)#

#60 + 8sqrt(2)#

Finally, use #4# as acommon factor to get

#60 + 8sqrt(2) = color(green)(4 * (15 + 2sqrt(2)))#