440. Найдите значение выражения, не пользуясь таблицей квадратов и калькулятором: 1) √20-66-330; 2) √3⁵⋅12³; 3) 2√18⋅3√30⋅5√15; 4) 6√10⋅√45⋅√50.

1. √20 ⋅ 66 ⋅ 330

Представим числа как произведения простых множителей: √ (2² ⋅ 5) ⋅ (2 ⋅ 3 ⋅ 11) ⋅ (2 ⋅ 3 ⋅ 5 ⋅ 11) = √ (2⁴ ⋅ 3² ⋅ 5² ⋅ 11²) = 2² ⋅ 3 ⋅ 5 ⋅ 11 = 4 ⋅ 3 ⋅ 5 ⋅ 11 = 660.

2. √3⁵ ⋅ 12³

Представим 12 как произведение простых множителей: √ 3⁵ ⋅ (2² ⋅ 3)³ = √ 3⁵ ⋅ 2⁶ ⋅ 3³ = √ 2⁶ ⋅ 3⁸ = 2³ ⋅ 3⁴ = 8 ⋅ 81 = 648.

3. 2√18 ⋅ 3√30 ⋅ 5√15

Запишем выражение в виде: 2 ⋅ 3 ⋅ 5 ⋅ √ (18 ⋅ 30 ⋅ 15) = 30 ⋅ √ (2 ⋅ 3² ⋅ 2 ⋅ 3 ⋅ 5 ⋅ 3 ⋅ 5) = 30 ⋅ √ (2² ⋅ 3⁴ ⋅ 5²) = 30 ⋅ 2 ⋅ 3² ⋅ 5 = 30 ⋅ 2 ⋅ 9 ⋅ 5 = 2700.

4. 6√10 ⋅ √45 ⋅ √50

Запишем выражение в виде: 6 ⋅ √ (10 ⋅ 45 ⋅ 50) = 6 ⋅ √ (2 ⋅ 5 ⋅ 3² ⋅ 5 ⋅ 2 ⋅ 5²) = 6 ⋅ √ (2² ⋅ 3² ⋅ 5⁴) = 6 ⋅ 2 ⋅ 3 ⋅ 5² = 6 ⋅ 2 ⋅ 3 ⋅ 25 = 900.