1. To rationalize the denominator, we need to multiply both the numerator and denominator by the conjugate of the denominator.
Let's start with the expression: 4√6/√30
The conjugate of √30 is -√30, so we will multiply both the numerator and denominator by -√30:
(4√6/√30) * (-√30/-√30)
Simplifying the expression, we get:
-4√6 * √30 / (-√30 * -√30)
Next, we multiply the terms within the numerator:
-4 * 6 * √30 * √30 / (-√30 * -√30)
Simplifying further:
-24 * 30 / (-√30 * -√30)
Multiplying the numbers:
-720 / (-√30 * -√30)
Finally, simplifying the square root:
-720 / (√30 * √30)
The square root of 30 is 5√2, so the expression becomes:
-720 / (5√2 * 5√2)
Simplifying:
-720 / (25 * 2)
-720 / 50
The final simplified expression is:
-14.4
2. We have the expression: (2√5 + 3√7)^2
To solve this, we'll use the formula (a + b)^2 = a^2 + 2ab + b^2.
Let's substitute a with 2√5 and b with 3√7:
(2√5 + 3√7)^2 = (2√5)^2 + 2(2√5)(3√7) + (3√7)^2
Simplifying the square terms:
(4*5) + 2(2√5)(3√7) + (9*7)
20 + 2(2√5)(3√7) + 63
Multiplying the numbers:
20 + 2(6√35) + 63
Next, we distribute the 2:
20 + 12√35 + 63
Adding the numbers:
83 + 12√35
Therefore, the simplifed expression is:
83 + 12√35