SIMPLIFY
1.(3+√2)/√2
2. 4/(1+√3)
3.(3+√5)(1-2√5)
4.(3+√8)/((2√3)-4)
Square root.
(3+√2)/√2 = (3√2 + 2)/2
4/(1+√3) * (√3-1)/(√3-1) = 4(√3-1)/(3-1) = 2(√3-1)
(3+√5)(1-2√5) = 3 -5√5 - 10 = -7-5√5
(3+√8)/((2√3)-4) * (2√3 + 4)/(2√3 + 4)
= (3+√8)(2√3 + 4)/(12-16)
= (6√3 + 2√24 - 12 - 4√8)/-4
= (12 + 8√2 - 6√3 - 4√6)/4
To simplify the given expressions, we'll follow these steps for each one:
1. Simplify (3+√2)/√2:
Step 1: Rationalize the denominator by multiplying the numerator and denominator by the conjugate of √2, which is -√2. This eliminates the square root in the denominator.
(3+√2)/√2 * (-√2)/(-√2) = (-3√2-2)/(2)
Therefore, the simplified expression is (-3√2-2)/(2).
2. Simplify 4/(1+√3):
Step 1: Rationalize the denominator by multiplying the numerator and denominator by the conjugate of (1+√3), which is (1-√3). This eliminates the square root in the denominator.
4/(1+√3) * (1-√3)/(1-√3) = (4-4√3)/(1-3)
Simplifying further, we get:
(4-4√3)/(-2) = -(2-2√3)
Therefore, the simplified expression is -(2-2√3).
3. Simplify (3+√5)(1-2√5):
Step 1: Apply the distributive property.
(3+√5)(1-2√5) = 3(1) + 3(-2√5) + √5(1) + √5(-2√5)
= 3 - 6√5 + √5 - 2√5√5
= 3 - 6√5 + √5 - 2√25
= 3 - 6√5 + √5 - 10
= (3 - 10) + (√5 - 6√5)
= -7 - 5√5
Therefore, the simplified expression is -7 - 5√5.
4. Simplify (3+√8)/((2√3)-4):
Step 1: Simplify the square root in the numerator.
√8 = √(4*2) = 2√2
Step 2: Simplify the square root in the denominator.
√3 = √(3) = √(1*3) = √(√1*√3) = √(√3)
Step 3: Apply the distributive property in the denominator.
(2√3) - 4 = 2√3 - 4
Therefore, the simplified expression is (2√2 + 3)/(2√3 - 4).