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ÖSYM \( 5^{x+1}=40 \) olduğuna göre, \[ \frac{25^{x}+16}{5^{x}+8} \] ifadesinin değeri kaçtır? A) 2 B) 3 C) 4 D) 5 E) 6

Ask by Parry Carlson. in Turkey
Feb 09,2025

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\[ \frac{25^{x}+16}{5^{x}+8} = 5 \] Cevap: 5 (D)

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Verilen denklem:   5^(x+1) = 40 Bu ifadeyi 5^x'in değerini bulmak için düzenleyelim:   5 · 5^x = 40 → 5^x = 40 / 5 = 8 İstenen ifade:   (25^x + 16) / (5^x + 8) 25^x ifadesini 5 tabanına göre ifade edelim:   25^x = (5^2)^x = 5^(2x) = (5^x)^2 = 8^2 = 64 Böylece ifademizin payı:   64 + 16 = 80 Payda:   5^x + 8 = 8 + 8 = 16 Son olarak:   80 / 16 = 5 Doğru cevap: 5 (D)

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Öncelikle eşitliği çözerek \(5^{x}\) değerini bulalım. Verilen ifade: \[ 5^{x+1} = 40 \] bu da demek oluyor ki: \[ 5^{x} \cdot 5 = 40 \implies 5^{x} = \frac{40}{5} = 8 \] Şimdi, ifademizde \(5^{x}\) yerine 8 yazalım: \[ \frac{25^{x}+16}{5^{x}+8} \] \(25^{x}\) ifadesini şu şekilde yazabiliriz: \[ 25^{x} = (5^{2})^{x} = (5^{x})^{2} = 8^{2} = 64 \] Bu durumda, ifadenin değeri şöyle olacak: \[ \frac{64 + 16}{8 + 8} = \frac{80}{16} = 5 \] Sonuç olarak, ifadenin değeri \(5\) olup, doğru cevap D seçeneğidir.

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