Simplify in radical form

2-√12 / 3+√5

Two minus radical twelve over three plus radical five. I believe I have to multiply by the conjugate but it’s coming out to a weird answer, could anyone solve and explain thanks.

parentheses help clarify ... (2-√12) / (3+√5)

... it's a different problem without them

multiplying by conjugate (FOIL) ... (6 - 2√5 - 3√12 + √60) / (9 - 5)

6 - 2√5 - 6√3 + 4√15) / 4 ... (3 - √5 - 3√3 + 2√15) / 2

I'll assume you meant

(2-√12) / (3+√5)
2(1-√3) / (3+√5)
2(1-√3)(3-√5) / (3+√5)(3-√5)
2(3-3√3-√5+√15)/(9-5)
(√3-1)(√5-3)/2

To simplify the expression (2 - √12)/(3 + √5) in radical form, we can indeed make use of the conjugate. The conjugate of the denominator 3 + √5 is 3 - √5.

To begin, we multiply both the numerator and the denominator by the conjugate:

((2 - √12)/(3 + √5)) * ((3 - √5)/(3 - √5))

Expanding this expression using the FOIL method, we get:

((2 * 3) + (2 * -√5) - (√12 * 3) - (√12 * -√5)) / ((3 * 3) + (3 * -√5) - (√5 * 3) - (√5 * -√5))

Simplifying this further, we have:

(6 - 2√5 - 3√12 + √60) / (9 - 3√5 - 3√5 + 5)

Now, let's simplify the radicals in the numerator:

6 - 2√5 - 3√(4 * 3) + √(4 * 15)
6 - 2√5 - 3 * 2√3 + 2√15

Notice that the term -2√5 and the term 2√15 are like terms, as are the constant terms 6 and 2√3. So, combining these like terms, we get:

(6 + 2√3) - 2√5 + 2√15

Finally, simplifying the expression in the denominator further:

9 - 3√5 - 3√5 + 5
14 - 6√5

Putting it all together, the simplified expression in radical form is:

(6 + 2√3 - 2√5 + 2√15) / (14 - 6√5)