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Answer
A expressão simplificada é \( \frac{4(\sqrt{5} + \sqrt{2})}{3} \).
Solution
Para racionalizar a expressão
4/(√5 − √2)
multiplicamos o numerador e o denominador pelo conjugado do denominador, que é (√5 + √2):
4/(√5 − √2) × (√5 + √2)/(√5 + √2) = [4(√5 + √2)]/[ (√5)² − (√2)² ].
Calculamos os quadrados:
(√5)² = 5 e (√2)² = 2.
Portanto, o denominador fica:
5 − 2 = 3.
Logo, a expressão se torna:
[4(√5 + √2)]/3.
Esse é o resultado final com o denominador racionalizado.
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Simplify this solution Beyond the Answer
To rationalize the denominator of \( \frac{4}{\sqrt{5}-\sqrt{2}} \), we can multiply both the numerator and the denominator by the conjugate of the denominator, which is \( \sqrt{5}+\sqrt{2} \). This gives us: \[ \frac{4(\sqrt{5}+\sqrt{2})}{(\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2})} \] The denominator simplifies as follows: \[ (\sqrt{5})^2 - (\sqrt{2})^2 = 5 - 2 = 3 \] Now we substitute back into the expression: \[ \frac{4(\sqrt{5}+\sqrt{2})}{3} \] So the rationalized form of \( \frac{4}{\sqrt{5}-\sqrt{2}} \) is: \[ \frac{4\sqrt{5}+4\sqrt{2}}{3} \] And there you have it—a rationalized expression that’s easy to work with!
