Question

Help! 1. На сколько sqrt(9+sqrt(65))*sqrt(9-sqrt(65) меньше 14? 2.Вычислите sqrt(3sqrt3(18))*sqrt6(96)

Answer 1

1)
$\sqrt{9 + \sqrt{65} } \times \sqrt{9 - \sqrt{65} } = \sqrt{(9 + \sqrt{65} )(9 - \sqrt{65}) } = \sqrt{{9}^{2} - {( \sqrt{65} )}^{2} } = \sqrt{81 - 65} = \sqrt{16} = 4 \\ 14 - 4 = 10$
Ответ: на 10.

2)
$\sqrt{3 \sqrt[3]{18} } \times \sqrt[6]{96} = \sqrt{ \sqrt[3]{27 \times 18} } \times \sqrt[6]{96} = \sqrt[6]{27 \times 18} \times \sqrt[6]{96} = \sqrt[6]{27 \times 18 \times 96} = \sqrt[6]{ {3}^{3} \times 2 \times {3}^{2} \times 3 \times 32 } = \sqrt[6]{ {3}^{6} \times 64} = \sqrt[6]{ {3}^{6} } \times \sqrt[6]{64} = 3 \times 2 = 6$
Ответ: 6

Answer 2

Sure, I can help you with those calculations.

1. To find the value of sqrt(9+sqrt(65))*sqrt(9-sqrt(65)) and determine how much it is less than 14, first, we need to evaluate the expression inside the square root.

sqrt(65) is approximately 8.06, so: sqrt(9+sqrt(65)) ≈ sqrt(9+8.06) ≈ sqrt(17.06) ≈ 4.12 sqrt(9-sqrt(65)) ≈ sqrt(9-8.06) ≈ sqrt(0.94) ≈ 0.97

Now, let's find the product of these two square roots: sqrt(9+sqrt(65)) * sqrt(9-sqrt(65)) ≈ 4.12 * 0.97 ≈ 3.9974 (approximately)

Now, let's determine how much this value is less than 14: 14 - 3.9974 ≈ 10.0026 (approximately)

So, sqrt(9+sqrt(65))*sqrt(9-sqrt(65)) is approximately 3.9974, and it is about 10.0026 less than 14.

2. To compute sqrt(3sqrt(3)(18))*sqrt(6)(96), let's break it down step by step:

Step 1: Simplify the expressions inside the square roots: sqrt(3 * sqrt(3) * 18) = sqrt(54) ≈ 7.35 sqrt(6 * 96) = sqrt(576) = 24

Step 2: Multiply the simplified expressions together: 7.35 * 24 ≈ 176.4

So, sqrt(3sqrt(3)(18))*sqrt(6)(96) is approximately 176.4.