Write the expression in simplified radical form.
2-2Square root 3 over
7+ square root 3
(2-2√3)/(7+√3)
recalling that (a+√b)(a-√b) = a^2-b,
(2-2√3)(7-√3)/(49-3)
(14-14√3-2√3+2*3)/46
(20-16√3)/46
(10-8√3)/23
To simplify the given expression, we need to rationalize the denominator. To do this, we multiply both the numerator and the denominator by the conjugate of the denominator, which is the expression obtained by changing the sign of the square root term.
The given expression is:
(2 - 2√3) / (7 + √3)
To rationalize the denominator, we multiply both numerator and denominator by (7 - √3), the conjugate of (7 + √3):
((2 - 2√3) * (7 - √3)) / ((7 + √3) * (7 - √3))
Expanding the numerator and the denominator, we get:
(14 - 2√3 - 14√3 + 6) / (49 - 3)
Combining like terms in the numerator, we have:
(20 - 16√3) / 46
This can be further simplified by dividing both the numerator and denominator by their greatest common factor, which in this case is 2:
(10 - 8√3) / 23
Therefore, the expression in simplified radical form is:
(10 - 8√3) / 23