7. Simplify. (Image: start-fraction 6 over start root 3 end root plus 2 end-fraction) (1 point)
(Image: Start Fraction 4 over Start Root 3 End Root End Fraction)
(Image: 12 minus 6 Start Root 3 End Root)
(Image: Start Root 3 End Root minus 2)
3
(Type "3" in the answer box)
Start by rationalizing the denominator of the first expression by multiplying both the numerator and denominator by the conjugate of the denominator: Start fraction 6 over Start root 3 end root plus 2 end fraction times Start fraction Start root 3 end root minus 2 over Start fraction Start root 3 end root minus 2 end fraction. This simplifies to (6√3-12)/(3-4) = -6(√3-2)/(-1) = 6(√3-2). Therefore, the simplified expression is Start fraction 4 over Start root 3 end root = 4/√3. To simplify the third expression, multiply by the conjugate: Start root 3 end root minus 2 times Start root 3 end root + 2 over Start root 3 end root + 2. This simplifies to (3-4)=-1. Therefore, the simplified expression is -1. Out of the given answer choices, only 3 matches this result.
7. Simplify. 6/√3 +2 (1 point)
4/√3
12-6√3
√3 -2
3
First, rationalize the denominator by multiplying both the numerator and denominator by √3 - 2:
(6/√3 + 2) * (√3 - 2)/(√3 - 2)
Simplifying this gives:
(6√3 - 12)/(3 - 4)
= -6(√3 - 2)/1
= -6√3 + 12
Out of the given answer choices, the closest equivalent is 12 - 6√3.
To simplify the expression (6/√3 + 2), we can follow these steps:
Step 1: Rationalize the denominator
Since the expression has a square root in the denominator, we want to eliminate it. To do this, we multiply both the numerator and the denominator by the conjugate of the denominator, which is √3.
(6/√3 + 2) * (√3/√3)
= (6√3 + 2√3) / (√3 * √3)
= (6√3 + 2√3) / 3
= (8√3) / 3
Step 2: Simplify the expression
We can now combine the like terms in the numerator.
(8√3) / 3
= (8/3) * √3
So, the simplified expression is (8/3)√3.