Soru

12)(sqrt (27)+sqrt (12))/(sqrt (45)-sqrt (5))cdot (sqrt (3))/(sqrt (5)) ipleminin A) 3 (1)/(2) (3)/(2) B (5)/(2)
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Perihan
Usta · 5 yıl öğretmeniUzman doğrulaması
Cevap
To solve the expression \frac{\sqrt{27} + \sqrt{12}}{\sqrt{45} - \sqrt{5}} \cdot \frac{\sqrt{3}}{\sqrt{5}}, we need to simplify each part of the expression step by step.
First, let's simplify the square roots in the numerator and denominator:
\sqrt{27} = \sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3}
\sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}
\sqrt{45} = \sqrt{9 \cdot 5} = \sqrt{9} \cdot \sqrt{5} = 3\sqrt{5}
Now substitute these simplified forms back into the original expression:
\frac{3\sqrt{3} + 2\sqrt{3}}{3\sqrt{5} - \sqrt{5}} \cdot \frac{\sqrt{3}}{\sqrt{5}}
Combine like terms in the numerator and denominator:
\frac{(3\sqrt{3} + 2\sqrt{3})}{(3\sqrt{5} - \sqrt{5})} = \frac{5\sqrt{3}}{2\sqrt{5}}
Next, multiply by \frac{\sqrt{3}}{\sqrt{5}}:
\frac{5\sqrt{3}}{2\sqrt{5}} \cdot \frac{\sqrt{3}}{\sqrt{5}} = \frac{5\sqrt{3} \cdot \sqrt{3}}{2\sqrt{5} \cdot \sqrt{5}} = \frac{5 \cdot 3}{2 \cdot 5} = \frac{15}{10} = \frac{3}{2}
Thus, the correct answer is:
C) \frac{3}{2}
First, let's simplify the square roots in the numerator and denominator:
\sqrt{27} = \sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3}
\sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}
\sqrt{45} = \sqrt{9 \cdot 5} = \sqrt{9} \cdot \sqrt{5} = 3\sqrt{5}
Now substitute these simplified forms back into the original expression:
\frac{3\sqrt{3} + 2\sqrt{3}}{3\sqrt{5} - \sqrt{5}} \cdot \frac{\sqrt{3}}{\sqrt{5}}
Combine like terms in the numerator and denominator:
\frac{(3\sqrt{3} + 2\sqrt{3})}{(3\sqrt{5} - \sqrt{5})} = \frac{5\sqrt{3}}{2\sqrt{5}}
Next, multiply by \frac{\sqrt{3}}{\sqrt{5}}:
\frac{5\sqrt{3}}{2\sqrt{5}} \cdot \frac{\sqrt{3}}{\sqrt{5}} = \frac{5\sqrt{3} \cdot \sqrt{3}}{2\sqrt{5} \cdot \sqrt{5}} = \frac{5 \cdot 3}{2 \cdot 5} = \frac{15}{10} = \frac{3}{2}
Thus, the correct answer is:
C) \frac{3}{2}
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