Example Question - algebraic expression simplification
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Simplification of a Mathematical Expression with Variables
Die dargestellte mathematische Ausdruck in dem Bild ist \( 13.6\sqrt{a^2 + b^2} + \frac{4}{8} \). Um diesen Ausdruck zu vereinfachen, betrachten wir die Termen einzeln. Der erste Term ist \( 13.6\sqrt{a^2 + b^2} \), welcher so bleibt, wie er ist, da keine weiteren Informationen über \( a \) oder \( b \) gegeben sind. Der zweite Term \( \frac{4}{8} \) kann vereinfacht werden, indem man den Bruch kürzt. Da 4 und 8 gemeinsame Teiler haben, nämlich 4, können wir sowohl den Zähler als auch den Nenner durch 4 teilen: \( \frac{4}{8} = \frac{4 \div 4}{8 \div 4} = \frac{1}{2} \). Damit ist der vereinfachte Ausdruck: \( 13.6\sqrt{a^2 + b^2} + 0.5 \). Es gibt hier nichts weiter zu vereinfachen, ohne zusätzliche Informationen über die Variablen \( a \) und \( b \).
Algebraic Expression Simplification
Para resolver la pregunta proporcionada en la imagen, parece que tenemos que simplificar la expresión algebraica. La expresión es: \[\frac{x^2 y^{-1/3}}{x^{-3} y^{1/2}} - \frac{7x^2}{2xy} \cdot \frac{10x^{-1}y}{x^{-1/2}y^1} + \frac{4x^5}{12x^4y}\] Empecemos por simplificar cada término individualmente: Para el primer término, utilizamos las propiedades de los exponentes para dividir las bases iguales restando los exponentes: \[\frac{x^2 y^{-1/3}}{x^{-3} y^{1/2}} = x^{2 - (-3)} y^{-\frac{1}{3} - \frac{1}{2}} = x^{2 + 3} y^{-\frac{1}{3} - \frac{2}{6}} = x^5 y^{-\frac{3}{6}} = x^5 y^{-\frac{1}{2}}\] Para el segundo término, primero simplificamos la fracción compuesta: \[\frac{7x^2}{2xy} \cdot \frac{10x^{-1}y}{x^{-1/2}y^1} = \frac{7x^2 \cdot 10x^{-1}y}{2xy \cdot x^{-1/2}y} = \frac{70x^{2-1}}{2x^{1-(-1/2)}y^{1+1}} = \frac{70x^{1}}{2x^{3/2}y^{2}}\] Dado que \(70/2 = 35\) y \(x^{1 - 3/2} = x^{-1/2}\), tenemos: \[\frac{70x^{1}}{2x^{3/2}y^{2}} = \frac{35}{y^{2}x^{1/2}}\] Para el tercer término, simplemente reducimos la fracción \(\frac{4x^5}{12x^4y}\): \[\frac{4x^5}{12x^4y} = \frac{4}{12} \cdot \frac{x^5}{x^4y} = \frac{1}{3} x^{5-4}y^{-1} = \frac{1}{3}xy^{-1}\] Ahora combinamos los términos simplificados: \[x^5 y^{-\frac{1}{2}} - \frac{35}{y^2x^{1/2}} + \frac{1}{3}xy^{-1}\] Este es la expresión simplificada de términos individuales de la expresión inicial en la imagen. Dada la complejidad de combinarlos en una expresión simple y común, podría ser más apropiado dejarlos en esta forma, a menos que tengamos un denominador común o una indicación de que necesitamos combinarlos de alguna manera específica. Si se necesita una simplificación adicional o una manipulación adicional, eso requeriría un contexto adicional o instrucciones específicas.
Algebraic Expression Simplification
The image shows an algebraic expression that requires simplification: (2√x - √y)(6√x + 5√y) This is a binomial multiplication problem, which we can solve using the FOIL method, where FOIL stands for "First, Outer, Inner, Last." This means we multiply the first terms in each binomial, then the outer terms, followed by the inner terms, and finally the last terms, combining like terms where possible. Here's how we apply FOIL to this problem: First: (2√x)*(6√x) = 12x Outer: (2√x)*(5√y) = 10√xy Inner: (-√y)*(6√x) = -6√xy Last: (-√y)*(5√y) = -5y Combining these results, we get: 12x + 10√xy - 6√xy - 5y Combine like terms: 12x + (10√xy - 6√xy) - 5y 12x + 4√xy - 5y The simplified form of the expression is: 12x + 4√xy - 5y
Simplifying an Algebraic Expression with Binomial Expansion
The image shows an algebraic expression to simplify: \[ 2^2 \cdot (2^{-\frac{1}{2}} + 1)^2 - (2^{\frac{3}{2}} + 1)^2 \] Let's simplify step by step: 1. Simplify the terms inside the parentheses first: \[ (2^{-\frac{1}{2}} + 1)^2 = ( \frac{1}{\sqrt{2}} + 1 )^2 \] \[ (2^{\frac{3}{2}} + 1)^2 = ( \sqrt{8} + 1 )^2 = (2\sqrt{2} + 1)^2 \] 2. Apply the binomial expansion to the squared terms: \[ ( \frac{1}{\sqrt{2}} + 1 )^2 = \left(\frac{1}{\sqrt{2}}\right)^2 + 2 \cdot \left(\frac{1}{\sqrt{2}}\right) \cdot 1 + 1^2 \] \[ (2\sqrt{2} + 1)^2 = (2\sqrt{2})^2 + 2 \cdot (2\sqrt{2}) \cdot 1 + 1^2 \] 3. Simplify each term: \[ \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \] \[ 2 \cdot \left(\frac{1}{\sqrt{2}}\right) = 2 \cdot \frac{1}{\sqrt{2}} = \sqrt{2} \] \[ (2\sqrt{2})^2 = 8 \] \[ 2 \cdot (2\sqrt{2}) = 4\sqrt{2} \] 4. Substitute back into the equation: \[ ( \frac{1}{2} + \sqrt{2} + 1 )^2 = ( \frac{3}{2} + \sqrt{2} )^2 \] \[ (8 + 4\sqrt{2} + 1 )^2 = (9 + 4\sqrt{2})^2 \] 5. Now multiply by the factor outside the parentheses in the original expression, which is \(2^2 = 4\), and subtract the squared terms: \[ 4 \cdot ( \frac{3}{2} + \sqrt{2} )^2 - (9 + 4\sqrt{2})^2 \] \[ = 4 \cdot (\frac{9}{4} + 3\sqrt{2} + 2) - (81 + 8\sqrt{2} + 16) \] \[ = (9 + 12\sqrt{2} + 8) - (81 + 8\sqrt{2} + 16) \] 6. Combine like terms: \[ 9 + 12\sqrt{2} + 8 - 81 - 8\sqrt{2} - 16 \] \[ = (9 + 8 - 81 - 16) + (12\sqrt{2} - 8\sqrt{2}) \] \[ = (-70) + 4\sqrt{2} \] So the simplified expression is: \[ -70 + 4\sqrt{2} \]
Algebraic Expression Simplification with Exponent Rules
This is an algebraic expression simplification problem involving exponent rules. We'll simplify each term by factoring out the common powers of 5 from the numerator and the denominator. Given expression: \( \frac{10 \times 5^{n+4} + 125 \times 5^{n+2}}{3 \times 5^{n+3} - 20 \times 5^{n+1}} \) Step 1: Factor out common powers of 5 from each term. Numerator: - Factor out \( 5^{n+2} \) from each term. \( 5^{n+2} (10 \times 5^2 + 125) = 5^{n+2} (10 \times 25 + 125) = 5^{n+2} (250 + 125) = 5^{n+2} \times 375 \) Denominator: - Factor out \( 5^{n+1} \) from each term. \( 5^{n+1} (3 \times 5^2 - 20) = 5^{n+1} (3 \times 25 - 20) = 5^{n+1} (75 - 20) = 5^{n+1} \times 55 \) Step 2: Simplify the expression with the factored out powers of 5. The simplified expression is: \( \frac{5^{n+2} \times 375}{5^{n+1} \times 55} \) Step 3: Cancel out the common power of 5 from the numerator and denominator. \( 5^{n+2} \) in the numerator and \( 5^{n+1} \) in the denominator have a common base, which means we can subtract the exponents. \( 5^{n+2 - (n+1)} = 5^1 = 5 \) So, the simplification yields: \( \frac{5 \times 375}{55} \) Step 4: Simplify the coefficients (375/55). Divide 375 by 55 to get: \( \frac{375}{55} = \frac{75}{11} \) So the final simplified expression is: \( \frac{5 \times 75}{11} \) Step 5: Multiply the 5 by 75. \( 5 \times 75 = 375 \) Therefore, the final simplified form of the expression is: \( \frac{375}{11} \) This fraction cannot be simplified further, as 375 and 11 do not have any common factors other than 1.
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