The equation x = 7 - 4√3 is a radical expression, and the value of the radical expression is [tex](x + \frac 1x)^2 = 3650401 -2107560\sqrt 3[/tex]
The equation is given as:
x = 7 - 4√3
So, we have:
[tex](x + \frac 1x)^2 = (7 - 4\sqrt 3 + \frac{1}{7 - 4\sqrt 3})^2[/tex]
Take the LCM
[tex](x + \frac 1x)^2 = (\frac{49 -56\sqrt3 + 48}{7 - 4\sqrt 3})^2[/tex]
Evaluate the like terms
[tex](x + \frac 1x)^2 = (\frac{97 -56\sqrt3 }{7 - 4\sqrt 3})^2[/tex]
Rationalize
[tex](x + \frac 1x)^2 = (\frac{(97 -56\sqrt3)(7 - 4\sqrt 3) }{49 -48})^2[/tex]
Evaluate the difference
[tex](x + \frac 1x)^2 = ((97 -56\sqrt3)(7 - 4\sqrt 3))^2[/tex]
Expand
[tex](x + \frac 1x)^2 = (679 -388\sqrt 3 -392\sqrt 3 + 672)^2[/tex]
Evaluate the like terms
[tex](x + \frac 1x)^2 = (1351 -780\sqrt 3 )^2[/tex]
Expand
[tex](x + \frac 1x)^2 = 1825201 +1825200 -2107560\sqrt 3[/tex]
[tex](x + \frac 1x)^2 = 3650401 -2107560\sqrt 3[/tex]
Hence, the value of the radical expression is [tex](x + \frac 1x)^2 = 3650401 -2107560\sqrt 3[/tex]
Read more about radical expressions at:
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