Example Question - exponential expression
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Determining the Value of x in an Exponential Expression
\[ \begin{align*} \frac{a^3 \cdot b^x \cdot c^4}{a^{1/2} \cdot b^{5/2} \cdot c^5} &= a^2 \cdot c^{-1} \\ a^{3 - \frac{1}{2}} \cdot b^{x - \frac{5}{2}} \cdot c^{4 - 5} &= a^2 \cdot c^{-1} \\ a^{\frac{5}{2}} \cdot b^{x - \frac{5}{2}} \cdot c^{-1} &= a^2 \cdot c^{-1} \\ \end{align*} \] Kita dapat memisahkan persamaan berdasarkan basis variabel a, b, dan c untuk membuatnya lebih mudah diselesaikan. \[ \begin{align*} a^{\frac{5}{2}} &= a^2 \\ b^{x - \frac{5}{2}} &= 1 \\ c^{-1} &= c^{-1} \\ \end{align*} \] Dari persamaan tersebut dapat disimpulkan bahwa: \[ \begin{align*} \frac{5}{2} &= 2 \quad \text{(Ini benar, tidak perlu dipecahkan lebih lanjut)} \\ x - \frac{5}{2} &= 0 \quad \text{(Ini yang akan kita selesaikan)} \\ \end{align*} \] \[ \begin{align*} x - \frac{5}{2} &= 0 \\ x &= \frac{5}{2} \\ \end{align*} \]
Solving an Exponential Expression
<p>Данное выражение: \( 4^{log_3{2}} \)</p> <p>По определению логарифма \( a^{log_a{b}} = b \), поэтому:</p> <p>\( 4^{log_3{2}} = 2^{log_3{4}} \)</p> <p>Используем свойство логарифмов \( log_a{b^n} = n \cdot log_a{b} \) и выразим \( log_3{4} \) как \( 2 \cdot log_3{2} \):</p> <p>\( 2^{log_3{4}} = 2^{2 \cdot log_3{2}} \)</p> <p>Тогда используем свойство экспонент: \( a^{m \cdot n} = (a^m)^n \):</p> <p>\( 2^{2 \cdot log_3{2}} = (2^{log_3{2}})^2 \)</p> <p>По определению логарифма возводим обратно:</p> <p>\( (3^{log_3{2}})^2 = 2^2 \)</p> <p>Таким образом, итоговый ответ:</p> <p>\( 2^2 = 4 \)</p>
Determining the Function Type Based on the Exponential Expression
<p>The function \( f(x) = 2 \cdot 5^x \) is an exponential function because it can be written in the form \( f(x) = ab^x \), where:</p> <p>\( a = 2 \) (the coefficient),</p> <p>\( b = 5 \) (the base), and</p> <p>\( x \) is the exponent.</p>
Determining the Value of an Exponential Expression Given a Linear Equation
Given: \( 6z - 4y = 8 \) We need to find the value of \( \frac{27z}{9y} \) First, solve for \( z \) in terms of \( y \): \( z = \frac{8}{6} + \frac{4y}{6} \) \( z = \frac{4}{3} + \frac{2y}{3} \) Now, substitute \( z \) into the expression \( \frac{27z}{9y} \): \( \frac{27(\frac{4}{3} + \frac{2y}{3})}{9y} \) \( \frac{27 \times \frac{4}{3}}{9y} + \frac{27 \times \frac{2y}{3}}{9y} \) \( \frac{36}{9y} + \frac{18y}{9y} \) \( \frac{36}{9y} + 2 \) Now, note that the second part of the expression is independent of \( y \), and the first part's numerator is not a multiple of \( y \) (since \( y \neq 0 \)), hence it cannot simplify to a constant. Therefore, the answer must be an integer that is 2 plus a positive number or 2 itself. Among the options given, only \( 9 \) satisfies this condition, since it's 7 more than 2. Thus the correct option is: \( A ) 9 \)
Evaluating Exponential Expression
The image displays a question asking for the answer to an expression: \(7^2\) To solve this, you simply need to calculate the square of 7, which means multiplying 7 by itself: \(7 \times 7 = 49\) So the answer to the question is 49.
Solving an Exponential Expression Step by Step
Por supuesto, vamos a resolver la expresión paso a paso. La expresión que se nos da es la siguiente: (-10)^4 * (-10) * (-10)^5 ÷ (-10)^3 Primero, vamos a simplificar las potencias y multiplicaciones. Sabemos que una base elevada a una potencia par siempre resulta en un número positivo, y una base negativa elevada a una potencia impar resulta en un número negativo. Así que: (-10)^4 = 10^4 = 10000, porque es una base a una potencia par. (-10)^5 = -10^5 = -100000, porque es una base a una potencia impar. Entonces multiplicamos los resultados junto con el (-10) adicional que está en medio. 10000 * (-10) * (-100000) = -100000 * (-100000) = 10000000000. Ahora, para la división que involucra potencias con la misma base, simplemente restamos los exponentes. Por lo tanto, dividir por (-10)^3 simplemente resta 3 al exponente de nuestra base (-10) actual, quedando así: 10000000000 ÷ 10^3 = 10000000000 ÷ 1000 = 10000000. Entonces el resultado final es 10000000.
Simplified Exponential Expression Calculation
The expression given in the image is 6 + 3^2. To simplify this expression, first calculate the exponent part (3^2), which means 3 multiplied by itself: 3^2 = 3 * 3 = 9 Now, add this result to 6: 6 + 9 = 15 Therefore, the simplified expression is 15.
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