problem #50

Perform the indicated division. Rationalize the denominator if necessary. Then simplify each radical expression.

(18+radical (567))/(9)

My answer: 2+ radical (7) or 2radical (7)

Problem # 52

(-9-radical(108))/(3)

My answer: -3-2radical(3)

Problem #54

(6-radical (20))/(2)

My answer: 3-radical (5)

To solve problems like these, you need to follow a few steps.

Step 1: Perform the division (if possible).

In problem #50, you have (18 + √567) / 9. To divide, you divide each term individually by 9. So, you have:

18 / 9 = 2
√567 / 9 = √63

Step 2: Simplify the radical.

To simplify the radical, you need to find the largest perfect square that can be factored out from the number under the radical. In this case, the largest perfect square factor of 63 is 9. So, you can rewrite √63 as √(9 * 7).

√(9 * 7) = √9 * √7 = 3√7

Therefore, the simplified form of √567 / 9 is (2 + 3√7).

In problem #52, you have (-9 - √108) / 3. Following the same steps, you get:

-9 / 3 = -3
√108 / 3 = √(36 * 3) / 3 = (√36 * √3) / 3 = 6√3 / 3 = 2√3

Therefore, the simplified form of (-9 - √108) / 3 is (-3 - 2√3).

In problem #54, you have (6 - √20) / 2. Following the same steps, you get:

6 / 2 = 3
√20 / 2 = √(4 * 5) / 2 = (√4 * √5) / 2 = 2√5 / 2 = √5

Therefore, the simplified form of (6 - √20) / 2 is (3 - √5).