Perform the indicated division. Rationalize the denominator if necessary. Then simplify each radical expression.
Problem #50
(18+radical(567))/(9)
my answer: 2+radical(7)
Problem # 52
(-9-radical (108))/(3)
My answer: -3-2radical (3)
yes on both.
To perform the indicated divisions and simplify the radical expressions, follow these steps:
Problem #50:
1. Start by dividing (18 + √567) by 9.
(18 + √567) / 9 = (18/9) + (√567/9)
2. Simplify the fraction 18/9.
18/9 = 2
3. Simplify the radical expression √567/9.
To simplify the radical expression, we need to find a perfect square that divides 567. In this case, 567 can be divided by 81 (which is 9^2), so we can rewrite √567 as √(81 * 7).
√567/9 = (√(81 * 7)) / 9
4. Apply the property of radicals, which states that √(a * b) = √a * √b.
(√(81 * 7)) / 9 = (√81 * √7) / 9
5. Simplify the radical √81.
√81 = 9
6. Substitute the values back into the expression.
(√(81 * 7)) / 9 = (9 * √7) / 9
7. Simplify the fraction (9 * √7) / 9.
(9 * √7) / 9 = √7
Therefore, the simplified expression is 2 + √7.
Problem #52:
1. Begin by dividing (-9 - √108) by 3.
(-9 - √108) / 3 = (-9/3) - (√108/3)
2. Simplify the fraction (-9/3).
-9/3 = -3
3. Simplify the radical expression √108/3.
To simplify the radical expression, we need to find a perfect square that divides 108. In this case, 108 can be divided by 36 (which is 6^2), so we can rewrite √108 as √(36 * 3).
√108/3 = (√(36 * 3))/3
4. Apply the property of radicals, which states that √(a * b) = √a * √b.
(√(36 * 3))/3 = (√36 * √3)/3
5. Simplify the radical √36.
√36 = 6
6. Substitute the values back into the expression.
(√(36 * 3))/3 = (6 * √3)/3
7. Simplify the fraction (6 * √3)/3.
(6 * √3)/3 = 2 * √3
Therefore, the simplified expression is -3 - 2√3.