Example Question - exponents rules
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Simplification of a Fraction with Exponents
\[ \left( \frac{4a^6b^5c^{-2}}{(2a^{-4}b^6c)^3} \right)^{-1} = \left( \frac{4a^6c^{-2}}{2^3a^{-12}b^{18}c^3} \right)^{-1} = \left( \frac{4a^{18}c^{-2}}{8b^{18}c^3} \right)^{-1} = \left( \frac{a^{18}c^{-2}}{2b^{18}c^3} \right)^{-1} = \frac{2b^{18}c^3}{a^{18}c^{-2}} \] \[ = 2b^{18}c^{3}a^{-18}c^{2} = 2b^{18}a^{-18}c^{5} \]
Properties of Exponents
Claro, te ayudaré a resolver la igualdad dada en la imagen utilizando las propiedades de los exponentes. La igualdad de la imagen es una propiedad de los exponentes que nos dice lo siguiente: La potencia de un producto es igual al producto de las potencias de los factores, es decir: \[(a \cdot b)^n = a^n \cdot b^n\] Y la potencia de un cociente es igual al cociente de las potencias del dividendo y el divisor, es decir: \[\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\] Ambas ecuaciones son expresiones generales de las propiedades de los exponentes para cualquier número real \(a\) y \(b\), y para cualquier exponente entero \(n\). Estas propiedades también se aplican a otros tipos de exponentes como fracciones, pero eso depende del contexto del problema. Las propiedades son útiles para simplificar expresiones algebraicas y facilitar la resolución de ecuaciones y otros problemas matemáticos. La imagen simplemente muestra estas dos propiedades sin ningún problema adicional que resolver. Por lo tanto, no se requiere una solución adicional más allá de entender y aplicar correctamente estas propiedades.
Simplifying Algebraic Expressions with Exponents
Sure, to simplify the expression provided in the image, we need to follow the rules of algebra and exponents. The expression in the image is: (10 × 5^n^4 + 125 × 5^n^2) / (3 × 5^n^3 + 20 × 5^n^1) Let's simplify the numerator and the denominator separately. First, we can factor out the greatest common factor in the numerator, which is 5^n^2: = 5^n^2 (10 × 5^n^2 + 125) / (3 × 5^n^3 + 20 × 5^n^1) And then we can do the same for the denominator, where the greatest common factor is 5^n^1: = 5^n^2 (10 × 5^n^2 + 125) / 5^n^1 (3 × 5^n^2 + 20) Now, let's simplify the terms inside the parentheses: Numerator: 10 × 5^n^2 + 125 can be simplified to 5^n^2 + 125. Since no further simplification can be done inside the parentheses (assuming the "+ 125" is not supposed to have a power of 5 associated with it), let's keep it as is. Denominator: 3 × 5^n^2 + 20 simplifies to 3 × 5^n^2 + 20. Again, assuming "20" does not have a power of 5 associated with it, there's no further simplification inside the parentheses. So now we have: = 5^n^2 (5^n^2 + 125) / 5^n^1 (3 × 5^n^2 + 20) Now, let's simplify across the numerator and denominator by canceling common factors where possible. We can cancel one 5^n^1 from both the numerator and the denominator: = 5^n (5^n^2 + 125) / (3 × 5^n^2 + 20) We cannot simplify further without knowing the value of n. If n is such that n^2, n^1, and the simple numerical terms 125 and 20 can be simplified further, we'd do that. Otherwise, this is the simplified form of the given expression. Please note that the superscripts in the original expression might be ambiguous due to the resolution and angle of the photo. If the powers are not correctly interpreted, please provide more information or adjust the expression accordingly.
Simplifying Algebraic Expression with Exponents
To simplify the expression given, we'll apply the rules of exponents and combine like terms. The expression is: \[ \frac{10x^{5n+4} + 125x^{5n+2}}{3x^{5n+3} - 20x^{5n+1}} \] Starting with the numerator: \[ 10x^{5n+4} + 125x^{5n+2} \] This expression cannot be simplified further because the terms are not like terms (they have different exponents). Now let's look at the denominator: \[ 3x^{5n+3} - 20x^{5n+1} \] Similarly, this expression cannot be simplified further because the terms are not like terms (they have different exponents). Since we cannot simplify the individual parts of the numerator or the denominator any further, we can only simplify if there's a common factor in both the numerator and the denominator. Given that the exponents in each term of the numerator are different from those in the denominator, there are no obvious factors that would cancel out. Therefore, the expression is already in its simplest form: \[ \frac{10x^{5n+4} + 125x^{5n+2}}{3x^{5n+3} - 20x^{5n+1}} \]
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