Under surd;
simplify the following.
1. 1 all over 2 root 3+ 5 that is 1/2√3+5.
2. 1 all over 1- root 3 all in bracket that is 1/(1-√3)^2.?
1 all over 2 root 3+ 5 that is 1/2√3+5 ---- strange, why not just say: 1/(2√3 + 5) ??
By "simplify" I will assume you mean "rationalize the denominator"
1/(2√3 + 5)
= 1/(2√3 + 5) * (2√3 - 5)/(2√3 - 5) <------using the conjugate
= (2√3 - 5)/(12 - 25)
= (5 - 2√3)/13
for the 2nd : 1/(1-√3)^2
first simplify (1-√3)^2
= 1 - 2√3 + 3
= 4 - 2√3
So 1/(1-√3)^2
= 1/(4 - 2√3)
proceed in the same way as I did for #1
1/[ 2 sqrt 3 + 5 ] * (2 sqrt 3 -5) / (2 sqrt 3 -5)
= (2 sqrt 3 - 5) / [(4*3)-25]
continue
To simplify these expressions, we need to rationalize the denominators. Rationalizing the denominator means eliminating surds (square roots) from the denominator.
Let's go through each expression one by one:
1. To simplify 1/(2√3 + 5):
- Multiply the numerator and denominator by the conjugate of the denominator, which is (2√3 - 5):
(1/(2√3 + 5)) * ((2√3 - 5)/(2√3 - 5))
- Applying the difference of squares: (a + b)(a - b) = a^2 - b^2, we can simplify the denominator:
(2√3)^2 - (5)^2 = 12 - 25 = -13
- Simplifying the entire expression:
(2√3 - 5)/(-13) = -(2√3 - 5)/13
So, 1/(2√3 + 5) simplifies to -(2√3 - 5)/13.
2. To simplify 1/(1 - √3)^2:
- First, expand the denominator using the formula: (a - b)^2 = a^2 - 2ab + b^2.
(1 - √3)^2 = (1^2 - 2(1)(√3) + (√3)^2)
= 1 - 2√3 + 3
= 4 - 2√3
- Simplifying the entire expression:
1/(1 - √3)^2 = 1/(4 - 2√3)
So, 1/(1 - √3)^2 simplifies to 1/(4 - 2√3).
Note: In some cases, you may be able to simplify further by rationalizing the denominator again or simplifying surds, but these are the simplified forms of the given expressions.