90, 91, 92

Краткие ответы:

• 90: 1) \(36\); 2) \(1{,}5\); 3) \(0{,}22\); 4) \(\dfrac{77}{12}=6\dfrac{5}{12}\); 5) \(144\); 6) \(2{,}88\).
• 91: 1) \(12\); 2) \(420\); 3) \(11\); 4) \(40\).
• 92: 1) \(45\); 2) \(120\); 3) \(4{,}8\); 4) \(104\).

Основное свойство

Для неотрицательных чисел:

\[ \sqrt{ab}=\sqrt a\cdot\sqrt b,\qquad \frac{\sqrt a}{\sqrt b}=\sqrt{\frac ab}. \]

Также:

\[ \sqrt{x^2}=|x|. \]

Упражнение 90

\[ 1)\quad \sqrt{81\cdot16}=9\cdot4=36. \]

\[ 2)\quad \sqrt{0{,}09\cdot25}=0{,}3\cdot5=1{,}5. \]

\[ 3)\quad \sqrt{0{,}01\cdot0{,}04\cdot121} =0{,}1\cdot0{,}2\cdot11=0{,}22. \]

\[ 4)\quad \sqrt{30\dfrac14\cdot\dfrac{49}{36}} = \sqrt{\dfrac{121}{4}\cdot\dfrac{49}{36}} = \sqrt{\left(\dfrac{77}{12}\right)^2} = \dfrac{77}{12}=6\dfrac5{12}. \]

\[ 5)\quad \sqrt{6^4\cdot4^2} = \sqrt{(6^2)^2\cdot4^2} =36\cdot4=144. \]

\[ 6)\quad \sqrt{(-2)^6\cdot0{,}3^4\cdot(-4)^2} = \left|(-2)^3\cdot0{,}3^2\cdot(-4)\right| \]

\[ =\left|(-8)\cdot0{,}09\cdot(-4)\right|=2{,}88. \]

Упражнение 91

\[ 1)\quad \sqrt{72}\cdot\sqrt2=\sqrt{144}=12. \]

\[ 2)\quad \sqrt{360}\cdot\sqrt{490} = \sqrt{36\cdot10}\cdot\sqrt{49\cdot10} =6\sqrt{10}\cdot7\sqrt{10}=420. \]

\[ 3)\quad \frac{\sqrt{242}}{\sqrt2} = \sqrt{\frac{242}{2}} =\sqrt{121}=11. \]

\[ 4)\quad \frac{\sqrt{40}}{\sqrt{0{,}025}} = \sqrt{\frac{40}{0{,}025}} =\sqrt{1600}=40. \]

Упражнение 92

\[ 1)\quad \sqrt{75\cdot27} = \sqrt{(25\cdot3)(9\cdot3)} =45. \]

\[ 2)\quad \sqrt{72\cdot200} = \sqrt{(36\cdot2)(100\cdot2)} =6\sqrt2\cdot10\sqrt2=120. \]

\[ 3)\quad \sqrt{1{,}6\cdot14{,}4} = \sqrt{23{,}04} =4{,}8, \]

так как \(4{,}8^2=23{,}04\).

\[ 4)\quad \sqrt{1690\cdot6{,}4} = \sqrt{10816}=104, \]

так как \(104^2=10816\).