Pergunta

f. (2sqrt (72))/(3)-(3sqrt (128))/(4)+5sqrt ((1)/(2))
Solução

4.0236 Voting

GonçaloMestre · Tutor por 5 anos
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To simplify the expression, we first simplify each term individually.<br /><br />1. Simplify \(\frac{2\sqrt{72}}{3}\):<br />\[<br />\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}<br />\]<br />\[<br />\frac{2 \times 6\sqrt{2}}{3} = \frac{12\sqrt{2}}{3} = 4\sqrt{2}<br />\]<br /><br />2. Simplify \(\frac{3\sqrt{128}}{4}\):<br />\[<br />\sqrt{128} = \sqrt{64 \times 2} = \sqrt{64} \times \sqrt{2} = 8\sqrt{2}<br />\]<br />\[<br />\frac{3 \times 8\sqrt{2}}{4} = \frac{24\sqrt{2}}{4} = 6\sqrt{2}<br />\]<br /><br />3. Simplify \(5\sqrt{\frac{1}{2}}\):<br />\[<br />\sqrt{\frac{1}{2}} = \frac{\sqrt{1}}{\sqrt{2}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}<br />\]<br />\[<br />5 \times \frac{\sqrt{2}}{2} = \frac{5\sqrt{2}}{2}<br />\]<br /><br />Now, combine all the simplified terms:<br />\[<br />4\sqrt{2} - 6\sqrt{2} + \frac{5\sqrt{2}}{2}<br />\]<br /><br />Combine like terms:<br />\[<br />(4\sqrt{2} - 6\sqrt{2}) + \frac{5\sqrt{2}}{2} = -2\sqrt{2} + \frac{5\sqrt{2}}{2}<br />\]<br /><br />Convert \(-2\sqrt{2}\) to a fraction with the same denominator:<br />\[<br />-2\sqrt{2} = \frac{-4\sqrt{2}}{2}<br />\]<br /><br />Now add the fractions:<br />\[<br />\frac{-4\sqrt{2}}{2} + \frac{5\sqrt{2}}{2} = \frac{-4\sqrt{2} + 5\sqrt{2}}{2} = \frac{\sqrt{2}}{2}<br />\]<br /><br />So, the simplified form of the given expression is:<br />\[<br />\boxed{\frac{\sqrt{2}}{2}}<br />\]
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