Example Question - expression simplification
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Algebraic Expression Simplification
Для решения этого задания необходимо упростить алгебраическое выражение, заданное в этом вопросе. Предположим, что изображение содержит выражение следующего вида: <p>\( b \left( a - b \right) - \left( b - a \right) a \)</p> Тогда решение будет следующим: <p>\( b(a - b) - (b - a)a \)</p> <p>\( = ba - b^2 - (ba - a^2) \)</p> <p>\( = ba - b^2 - ba + a^2 \)</p> <p>\( = a^2 - b^2 \)</p> Итак, упрощенное выражение \( a^2 - b^2 \).
Algebraic Expression Simplification
\[ \begin{array}{l} \text{Given: } y + \frac{1}{y} = b \\ \text{Squaring both sides: } \\ \left( y + \frac{1}{y} \right)^2 = b^2 \\ y^2 + 2 \cdot y \cdot \frac{1}{y} + \left( \frac{1}{y} \right)^2 = b^2 \\ y^2 + 2 + \frac{1}{y^2} = b^2 \\ \text{Subtract 2 from both sides: } \\ y^2 + \frac{1}{y^2} = b^2 - 2 \end{array} \]
Algebraic Expression Simplification
<p>Para resolver la expresión algebraica dada, se debe aplicar la propiedad distributiva (ley distributiva).</p> <p>La propiedad distributiva establece que \(a(b+c) = ab + ac\). En este caso, tendremos términos similares con \(x^3 \times x^3\) y \(x^3 \times (-x^7)\).</p> <p>Para simplificar, seguimos estos pasos:</p> <p>\((x^3)(x^3 - x^7) = x^3 \times x^3 - x^3 \times x^7\)</p> <p>Utilizamos la ley de los exponentes que dice que \(x^a \times x^b = x^{a+b}\):</p> <p>\(x^3 \times x^3 = x^{3+3} = x^6\)</p> <p>\(x^3 \times (-x^7) = -x^{3+7} = -x^{10}\)</p> <p>Combinando estos términos, la expresión se simplifica a:</p> <p>\(x^6 - x^{10}\)</p>
Step-by-Step Expression Simplification
Давайте решим данное выражение шаг за шагом. Помните, что приоритет операций следующий: сначала выполняем операции в скобках, затем умножение и деление, и в последнюю очередь сложение и вычитание. Выражение в скобках: \[ \frac{5}{9} - \frac{4}{9} = \frac{5-4}{9} = \frac{1}{9} \] Затем во вторых скобках решим деление: \[ \frac{3}{8} \div 3 = \frac{3}{8} \div \frac{3}{1} = \frac{3}{8} \times \frac{1}{3} = \frac{3 \times 1}{8 \times 3} = \frac{1}{8} \] Далее сложим результаты в скобках: \[ \frac{1}{9} + \frac{1}{8} = \frac{8}{72} + \frac{9}{72} = \frac{17}{72} \] Теперь умножим 0.9 на полученный результат: \[ 0.9 \times \frac{17}{72} = \frac{9}{10} \times \frac{17}{72} = \frac{9 \times 17}{10 \times 72} = \frac{153}{720} \] Упростим дробь, разделив числитель и знаменатель на их наибольший общий делитель, который равен 9: \[ \frac{153}{720} = \frac{17 \times 9}{80 \times 9} = \frac{17}{80} \] Итак, итоговый результат выражения равен \(\frac{17}{80}\).
Arithmetic Calculation and Evaluation
To solve the expression given in the image, follow the order of operations, PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right): Given expression: 7 + 3 × 2^4 ÷ 8 - (9 ÷ 6 - 3)^2 Let's simplify the expression step by step: 1. Solve the exponentiation (2^4): 7 + 3 × 16 ÷ 8 - (9 ÷ 6 - 3)^2 2. Perform the division inside the parentheses (9 ÷ 6): 7 + 3 × 16 ÷ 8 - (1.5 - 3)^2 3. Solve the subtraction inside the parentheses (1.5 - 3): 7 + 3 × 16 ÷ 8 - (-1.5)^2 4. Solve the exponentiation (-1.5)^2 (square of -1.5): 7 + 3 × 16 ÷ 8 - 2.25 5. Perform the multiplication (3 × 16) and the division by 8: 7 + 48 ÷ 8 - 2.25 6. Perform the division (48 ÷ 8): 7 + 6 - 2.25 7. Add 7 and 6: 13 - 2.25 8. Subtract 2.25 from 13: 13 - 2.25 = 10.75 However, none of the answer choices match this calculation. It seems we have faced an arithmetic mistake. Let us go through the calculation again more carefully. 1. Solve the exponent (2^4 = 16): 7 + 3 × 16 ÷ 8 - (9 ÷ 6 - 3)^2 2. Carry out the division (16 ÷ 8 = 2): 7 + 3 × 2 - (9 ÷ 6 - 3)^2 3. Perform any multiplications (3 × 2 = 6): 7 + 6 - (9 ÷ 6 - 3)^2 4. Solve the division inside the parentheses (9 ÷ 6 = 1.5): 7 + 6 - (1.5 - 3)^2 5. Perform the subtraction inside the parentheses (1.5 - 3 = -1.5): 7 + 6 - (-1.5)^2 6. Compute the square (-1.5 × -1.5 = 2.25): 7 + 6 - 2.25 7. Add the 7 and 6: 13 - 2.25 8. Finally, subtract 2.25 from 13: 13 - 2.25 = 10.75 After following each step carefully, the result is indeed 10.75, which does not appear to be one of the options provided. There may be a mistake in the answer choices or a misinterpretation of the question. If we consider the initial expression again and assume that order of division and multiplication as per the provided image: 7 + (3 × 2^4 ÷ 8) - (9 ÷ 6 - 3)^2 We may consider performing the division in the term (3 × 2^4 ÷ 8) following the left-to-right rule. In that case, we divide first and then multiply: 1. Solve the exponent (2^4 = 16): 7 + (3 × 16 ÷ 8) - (9 ÷ 6 - 3)^2 2. Perform the division first according to the left-to-right rule of division and multiplication (16 ÷ 8 = 2): 7 + (3 × 2) - (9 ÷ 6 - 3)^2 3. Multiply (3 × 2 = 6): 7 + 6 - (9 ÷ 6 - 3)^2 4. Solve within the parentheses for division and subtraction (9 ÷ 6 = 1.5, then 1.5 - 3 = -1.5): 7 + 6 - (-1.5)^2 5. Calculate the square of -1.5 (-1.5 × -1.5 = 2.25): 7 + 6 - 2.25 6. Add 7 and 6: 13 - 2.25 7. Subtract 2.25 from 13: 13 - 2.25 = 10.75 Once again, the result is 10.75. The question may contain a mistake, as none of the provided answer choices match the correct calculation. Could you please double-check the problem or the answer choices?
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