Example Question - complex fraction
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Simplifying a Complex Fraction with Polynomials
<p>Первый шаг - упростить числитель и знаменатель дроби, если это возможно.</p> <p>\[ \frac{{x^5 - 5x^2 + 8x + 4}}{{x^3 - x^2}} + \frac{{x + 2}}{{x^2}} \]</p> <p>Знаменатель первой дроби можно упростить, вынеся x^2:</p> <p>\[ x^2 \left( \frac{{x^3 - 5x + 8}}{{x}} + 4 \right) \]</p> <p>Знаменатель первой дроби теперь \( x \), а второй \( x^2 \).</p> <p>Чтобы сложить эти дроби, нужно привести их к общему знаменателю \( x^2 \):</p> <p>\[ \frac{{x(x^3 - 5x + 8) + 4x^2}}{{x^2}} + \frac{{x + 2}}{{x^2}} \]</p> <p>Теперь складываем числители двух дробей:</p> <p>\[ \frac{{x^4 - 5x^2 + 8x + 4x^2 + x + 2}}{{x^2}} \]</p> <p>Упрощаем выражение в числителе:</p> <p>\[ \frac{{x^4 - x^2 + 8x + 2}}{{x^2}} \]</p> <p>Финальный шаг – разделить каждый член в числителе на \( x^2 \):</p> <p>\[ x^2 - 1 + 8 \cdot \frac{x}{{x^2}} + \frac{2}{{x^2}} \]</p> <p>Результат:</p> <p>\[ x^2 - 1 + \frac{8}{x} + \frac{2}{{x^2}} \]</p>
Complex Fraction Simplification
La solución se realiza simplificando la fracción compleja paso por paso. <p>\begin{align*} &\frac{\frac{2}{3} + \frac{4}{7}}{\frac{9}{28} - \frac{1}{7}}\\\\ &= \frac{\frac{2 \cdot 7 + 4 \cdot 3}{3 \cdot 7}}{\frac{9 - 4}{28}} \text{ (encontrando un denominador común)}\\\\ &= \frac{\frac{14 + 12}{21}}{\frac{5}{28}} \text{ (simplificando los numeradores)}\\\\ &= \frac{\frac{26}{21}}{\frac{5}{28}} \text{ (simplificando la suma)}\\\\ &= \frac{26}{21} \times \frac{28}{5} \text{ (multiplicando por el inverso de la fracción del denominador)}\\\\ &= \frac{26 \cdot 28}{21 \cdot 5} \text{ (multiplicando numeradores y denominadores)}\\\\ &= \frac{728}{105} \text{ (realizando la multiplicación)}\\\\ &= \frac{56 \cdot 13}{21 \cdot 5} \text{ (descomponiendo 728 como } 56 \times 13 \text{)}\\\\ &= \frac{13 \cdot 56}{5 \cdot 21} \text{ (reorganizando los factores)}\\\\ &= \frac{13 \cdot 8}{5 \cdot 3} \text{ (simplificando } 56 / 21 \text{ a } 8 / 3 \text{)}\\\\ &= \frac{104}{15} \text{ (realizando la multiplicación final)}. \end{align*}</p>
Rationalizing the Denominator of a Complex Fraction
To express the given expression in the form \( q + r \sqrt{s} \), where \( q, r, \) and \( s \) are rational numbers, we need to rationalize the denominator. The given expression is: \[ \frac{2\sqrt{5} + 5\sqrt{2}}{2\sqrt{5} - 5\sqrt{2}} \] First, we can multiply the numerator and the denominator by the conjugate of the denominator to eliminate the square roots in the denominator. The conjugate of \(2\sqrt{5} - 5\sqrt{2}\) is \(2\sqrt{5} + 5\sqrt{2}\). Multiplying both the numerator and the denominator by the conjugate, we get: \[ \frac{(2\sqrt{5} + 5\sqrt{2})(2\sqrt{5} + 5\sqrt{2})}{(2\sqrt{5} - 5\sqrt{2})(2\sqrt{5} + 5\sqrt{2})} \] Now expand both the numerator and the denominator: The numerator, when multiplied out, becomes: \[ (2\sqrt{5} \cdot 2\sqrt{5}) + (2\sqrt{5} \cdot 5\sqrt{2}) + (5\sqrt{2} \cdot 2\sqrt{5}) + (5\sqrt{2} \cdot 5\sqrt{2}) \] \[ = 4 \cdot 5 + 2 \cdot 5 \cdot \sqrt{10} + 5 \cdot 2 \cdot \sqrt{10} + 25 \cdot 2 \] \[ = 20 + 10\sqrt{10} + 10\sqrt{10} + 50 \] \[ = 70 + 20\sqrt{10} \] The denominator, when multiplied out, becomes a difference of squares: \[ (2\sqrt{5})^2 - (5\sqrt{2})^2 \] \[ = 4 \cdot 5 - 25 \cdot 2 \] \[ = 20 - 50 \] \[ = -30 \] Combining the expanded numerator and the expanded denominator, we have: \[ \frac{70 + 20\sqrt{10}}{-30} \] Now, simplify this by dividing both terms in the numerator by the denominator: \[ \frac{70}{-30} + \frac{20\sqrt{10}}{-30} \] \[ = -\frac{7}{3} - \frac{2\sqrt{10}}{3} \] So the final answer, expressed in the form \( q + r \sqrt{s} \), is: \[ q = -\frac{7}{3}, \quad r = -\frac{2}{3}, \quad s = 10 \]
Solving a Complex Fraction to Standard Form
The provided expression is a complex fraction: \[ \frac{8 - i}{3 - 2i} \] To write this in the standard form \( a + bi \), where \( a \) and \( b \) are real numbers, you must remove the imaginary part from the denominator. You can do this by multiplying both the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of \(3 - 2i\) is \(3 + 2i\). Multiplying the numerator and the denominator by \(3 + 2i\) gives us: \[ \frac{8 - i}{3 - 2i} \times \frac{3 + 2i}{3 + 2i} \] This multiplication will give us: \[ \frac{(8 - i)(3 + 2i)}{(3 - 2i)(3 + 2i)} \] Expand the numerator and the denominator: \[ \frac{24 + 16i - 3i - 2i^2}{9 - 6i + 6i - 4i^2} \] Since \(i^2 = -1\), this simplifies to: \[ \frac{24 + 13i - 2(-1)}{9 - 4(-1)} \] \[ \frac{24 + 13i + 2}{9 + 4} \] \[ \frac{26 + 13i}{13} \] \[ 2 + i \] So in the form \( a + bi \), \( a = 2 \) and \( b = 1 \). Therefore, the value of \( a \) is 2, which corresponds to option A.
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