divide and simplify the radical

(3+ sqrt18)/4 sqrt12

I'm not sure how to conjugate the denominator?

repartir 152 entre A,B,C de modo que la parte de B sea 8 menos que el duplo de la de A y 32 mas que la de C

(3+√18) / 4√12

= (3+3√2) / 8√3
= 3√3 (1+√2) / 24
= (1+√2)√3 / 8
= (√3+√6)/8

or several other ways to express it

B = 2A-8

B = C+32
A+B+C = 152

C = B-32 = 2A-8-32

A + 2A-8 + 2A-40 = 152
5A - 48 = 152
5A = 200
A=40
B=72
C=40

To divide and simplify the given expression (3+√18) / (4√12), we need to rationalize the denominator to eliminate any radicals.

Rationalizing the denominator involves multiplying both the numerator and denominator by a suitable expression that eliminates the radical in the denominator. In this case, we will multiply by the conjugate of the denominator to achieve that.

The conjugate of the denominator, 4√12, is obtained by changing the sign of the radical. So, the conjugate is -4√12.

To rationalize the denominator and divide the expression, we follow these steps:

Step 1: Multiply both the numerator and denominator by the conjugate of the denominator.

(3+√18) / (4√12) * (-4√12) / (-4√12)

Step 2: Simplify the numerator by applying the distributive property.

(-4√12 * 3) + (-4√12 * √18)

-12√12 - 4√(12*18)

-12√12 - 4√216

Step 3: Simplify each radical term.

-12√12 - 4√(36*6)

-12√12 - 4√36√6

-12√12 - 4 * 6√6

-12√12 - 24√6

Step 4: Combine like terms (if any).

-12√12 - 24√6 (There are no like terms to combine in this case.)

And that's the simplified form of the expression (3+√18) / (4√12).