1. Perform indicated operations:
To simplify the expression 3√3 + 2√27 - √12, you correctly broke down the radicals into their simplest form:
3√3 + 2√(3*9) - √(2*6)
= 3√3 + 2√(3^2) - √(2^2*3)
= 3√3 + 2*3√3 - 2√3
= 3√3 + 6√3 - 2√3
= 7√3
So your final answer is 7√3.
2. Simplify:
To simplify the expression √(49x^12y^4z^8), you correctly applied the property of radicals that says √(ab) = √a * √b:
√(49x^12y^4z^8) = √(7^2 * x^12 * y^4 * z^8)
= 7 * x^6 * y^2 * z^4
So your final answer is 7x^6y^2z^4.
3. Multiply:
In your first attempt to multiply (8√6 + 3√2)(4√6 - 5√2), you made a mistake in distributing √36 and √4 square roots. Here's the correct step-by-step process:
(8√6 + 3√2)(4√6 - 5√2)
= 8√6 * 4√6 + 8√6 * (-5√2) + 3√2 * 4√6 + 3√2 * (-5√2)
= 32√36 - 40√12 + 12√12 - 15√4
Now, simplify each radical:
32√36 = 32√(6^2) = 32 * 6 = 192
40√12 = 40√(2^2 * 3) = 40 * 2√3 = 80√3
12√12 = 12√(2^2 * 3) = 12 * 2√3 = 24√3
15√4 = 15√(2^2) = 15 * 2 = 30
Combining these terms, you get:
192 - 40√3 + 24√3 - 30 = 162 - 16√3
So your final answer is 162 - 16√3.
Rationalizing the Denominator:
To rationalize the expression 5/(√3 + √5), you correctly multiplied both the numerator and the denominator by the conjugate of the denominator, which is √3 - √5:
5(√3 - √5)/(√3 + √5)(√3 - √5)
= 5(√3 - √5)/(3 - 5)
= 5(√3 - √5)/(-2)
So your final answer is (5√3 - 5√5)/(-2).