Example Question - solving expressions
Here are examples of questions we've helped users solve.
Solving the Expression with Distribution and Expansion
To solve the expression given in the image: 1. Distribute the \( 3p \) across \( (p - q) \): \[ 3p \cdot p + 3p \cdot (-q) \] 2. Expand the square \( (2p - q)^2 \) using the FOIL (First, Outer, Inner, Last) method: \[ (2p - q)(2p - q) \] \[ 2p \cdot 2p + 2p \cdot (-q) + (-q) \cdot 2p + (-q) \cdot (-q) \] Combine the above expansions: \[ 3p^2 - 3pq - (4p^2 - 2pq - 2pq + q^2) \] 3. Expand the negative sign into the second parenthesis: \[ 3p^2 - 3pq - 4p^2 + 2pq + 2pq - q^2 \] 4. Combine like terms: \[ 3p^2 - 4p^2 - 3pq + 2pq + 2pq - q^2 \] \[ -p^2 + pq - q^2 \] Therefore: \[ 3p(p - q) - (2p - q)^2 = -p^2 + pq - q^2 \]
Solving Mathematical Expressions with PEMDAS
To solve the mathematical expression in the image correctly, you need to follow the order of operations, which is often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right). The expression given is: \[ 8 \div 2(2+2) \] First, solve the operation inside the parentheses: \[ 2+2 = 4 \] Then the expression becomes: \[ 8 \div 2(4) \] \[ 8 \div 2 \times 4 \] According to PEMDAS, division and multiplication should be performed from left to right: \[ 4 \times 4 = 16 \] Therefore, the correct result of the expression is: \[ 16 \]
Solving a 3x3 Magic Square Problem with Expressions
Claro, analicemos el problema paso a paso. El problema trata sobre un cuadrado mágico 3x3 donde debemos ubicar los números 4, 6 y 8 de tal manera que la suma de cada fila, cada columna y cada diagonal sea igual a 18. Tenemos que identificar los valores de x, y y z y luego encontrar el valor de la expresión \(3x + y - 2z\). Para resolver el cuadrado mágico, usemos primero el hecho de que cada fila, columna y diagonal suman 18 y veamos cómo podemos distribuir los números. Podemos observar que cada número (4, 6, 8) se usa tres veces en el cuadrado (una vez en cada fila y una vez en cada columna). Esto significa que la suma total de los números en el cuadrado mágico será \(3(4 + 6 + 8)\), lo que simplificado es \(3 \times 18\), igual a 54. Teniendo esto en cuenta, podemos calcular la suma de una fila (o columna o diagonal). Puesto que hay 3 de ellas, cada fila (o columna o diagonal) debe sumar \(54 \div 3\), lo cual es igual a 18. Además, el número central y se repite en cada una de las filas, columnas y diagonales, lo que significa que y debe ser 6, puesto que es el único número que, al multiplicarse por 3, da 18. Ahora tendremos que ubicar los números x y z de forma que las filas, columnas y diagonales sumen también 18. El cuadrado mágico quedaría de la siguiente forma: ``` x | y | z --------- y | y | y --------- z | y | x ``` Si escogemos x y z como 4 y 8 de alguna manera, podemos establecer una ecuación para una fila o columna y resolver para x y z. Tomando, por ejemplo, la primera fila: x + y + z = 4 + 6 + 8 = 18 Tenemos que las posibles combinaciones de x y z son (4, 8) o (8, 4), verifiquemos si las esquinas suman 18: Para x = 4 y z = 8: esquina superior izquierda (x) + centro (y) + esquina inferior derecha (x) = 4 + 6 + 4 = 14 (no es 18) Para x = 8 y z = 4: esquina superior izquierda (x) + centro (y) + esquina inferior derecha (x) = 8 + 6 + 8 = 22 (no es 18) De modo que la distribución correcta sería x = 4 en las esquinas y z = 8 en las posiciones medias de los bordes. Ahora que sabemos que x es 4 y z es 8, podemos encontrar el valor de la expresión: \(3x + y - 2z = 3(4) + 6 - 2(8) = 12 + 6 - 16 = 18 - 16 = 2.\) El valor de \(3x + y - 2z\) es 2. Sin embargo, al observar las opciones, notamos que el valor 2 no está listado, lo cual indica que posiblemente hubo un error en el razonamiento. Revisemos de nuevo la lógica del problema. El error estuvo en asignar x = 4 en las esquinas y z = 8 en las posiciones medias de los bordes; no cumplen con las sumas de las diagonales que deben ser 18. Reasignemos correctamente entonces: si y = 6, colocado al centro, las esquinas deben ser 4 y 8 también, pero no al azar. Si la esquina superior izquierda es 4, la esquina inferior derecha debe ser 8 para que sumen 18 con el centro. Lo mismo sucede con las esquinas restantes. El cuadrado mágico correcto es: ``` 4 | 6 | 8 --------- 8 | 6 | 4 --------- 4 | 6 | 8 ``` Ahora, calculamos la expresión: \(3x + y - 2z = 3(4) + 6 - 2(8) = 12 + 6 - 16 = 18 - 16 = 2.\) Esta vez, el cálculo fue correcto, pero parece que hubo un malentendido con la asignación a las letras x, y y z. Asumiendo que queremos decir que x es la casilla superior izquierda, que es 4, y es el centro, que es 6, y z es la casilla superior derecha, que es 8, entonces el cálculo quedaría: \(3x + y - 2z = 3(4) + 6 - 2(8) = 12 + 6 - 16 = 18 - 16 = 2.\) Sin embargo, al revisar las opciones proporcionadas, el valor de 2 no está presente. Esto significa que se ha cometido un error nuevamente en la interpretación de x, y, y z en la fórmula proporcionada. Si asumimos que la x, y y z en \(3x + y - 2z\) corresponden exactamente a los lugares que ocupan en el cuadrado mágico del problema, es decir, x = 8, y = 6 y z = 4, entonces la expresión sería: \(3x + y - 2z = 3(8) + 6 - 2(4) = 24 + 6 - 8 = 30 - 8 = 22.\) La opción correcta es la que coincide con el número 22, que es la opción E) 22.
Incorrect Alternative Expressions for B√3
To solve this question, we need to find an alternative expression for \( B\sqrt{3} \). Option A is \( \sqrt{B} \times \sqrt{3} \), which simplifies to \( \sqrt{B \times 3} \) or \( \sqrt{3B} \), not \( B\sqrt{3} \). Option B is \( \sqrt{B} \times \sqrt{3} \), which is identical to Option A and thus also incorrect. Option C is \( \frac{\sqrt{3}}{\sqrt{B}} \), this isn't equivalent to \( B\sqrt{3} \) either. If anything, it represents \( \sqrt{3}/\sqrt{B} \) which is \( \sqrt{3/B} \), not the expression we are looking for. Option D is \( \frac{\sqrt{B}}{\sqrt{3}} \), and this simplifies to \( \sqrt{B/3} \), which is also not equivalent to \( B\sqrt{3} \). However, from the options provided, none match \( B\sqrt{3} \). Each option represents a different expression upon simplification. Thus, it seems there might be an error as none of the given options is another name for \( B\sqrt{3} \).
Solving Expressions with Given Values of x
Dựa trên hình ảnh bạn cung cấp, chúng ta sẽ giải Bài 3, bài toán về việc tìm giá trị của x trong hai biểu thức sau: a) 3x - 7 trái với x = 2. Đầu tiên, chúng ta sẽ thay x bằng 2 vào biểu thức 3x - 7 để tìm giá trị của biểu thức. 3x - 7 = 3(2) - 7 = 6 - 7 = -1. Như vậy, giá trị của biểu thức 3x - 7 khi x = 2 là -1. b) x^2 - 2x^2 + 3(x + 1) trái với x = -1. Chúng ta sẽ làm tương tự bằng cách thay x = -1 vào biểu thức x^2 - 2x^2 + 3(x + 1): x^2 - 2x^2 + 3(x + 1) = (-1)^2 - 2(-1)^2 + 3(-1 + 1) = 1 - 2(1) + 3(0) = 1 - 2 + 0 = -1. Vậy, giá trị của biểu thức x^2 - 2x^2 + 3(x + 1) khi x = -1 là -1.
Solving Expression with Fraction Exponents in Vietnamese
Tôi sẽ giúp bạn giải câu 1(a) bằng tiếng Việt: Đề bài yêu cầu rút gọn biểu thức: \[ A = \left(\frac{2}{3}\right)^{-2} \cdot \left(\frac{-3}{2}\right)^{3} \cdot \left(\frac{1}{6}\right)^{0} \] Bây giờ, chúng ta sẽ tiến hành rút gọn từng phần của biểu thức: 1. \(\left(\frac{2}{3}\right)^{-2}\) là nghịch đảo của \(\left(\frac{2}{3}\right)^{2}\), vậy ta có: \[\left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^{2} = \frac{3^2}{2^2} = \frac{9}{4}\] 2. \(\left(\frac{-3}{2}\right)^{3}\) là lập phương của \(\frac{-3}{2}\), nên ta có: \[\left(\frac{-3}{2}\right)^{3} = \left(\frac{-3}{2}\right) \cdot \left(\frac{-3}{2}\right) \cdot \left(\frac{-3}{2}\right) = \frac{-27}{8}\] 3. \(\left(\frac{1}{6}\right)^{0}\) mọi số mũ 0 đều bằng 1, nên ta có: \[\left(\frac{1}{6}\right)^{0} = 1\] Giờ, nhân tất cả các kết quả với nhau, ta có: \[A = \frac{9}{4} \cdot \frac{-27}{8} \cdot 1 = \frac{9 \cdot (-27)}{4 \cdot 8} = \frac{-243}{32}\] Vậy kết quả của biểu thức A là \(\frac{-243}{32}\). Hy vọng điều này giúp bạn giải quyết bài toán. Nếu bạn cần giúp đỡ với các phần khác của bài tập, đừng ngần ngại yêu cầu thêm giải đáp.
Solving an Expression for a Given Value
To solve the expression given in the image for x = -1, we will plug in x = -1 and simplify: Expression: 8(x+1)^2 - 2(x^2 + 6x) Plugging in x = -1 gives: 8(-1 + 1)^2 - 2((-1)^2 + 6(-1)) = 8(0)^2 - 2(1 - 6) = 0 - 2(-5) = 0 + 10 = 10 Therefore, the correct answer is not listed among the choices provided in the image. If this is a question from a test, there might be an error in the question or the answer choices.
Solving Expression for Specific Value
To solve the given expression for x = 11, you simply substitute the value of x into the expression and then perform the arithmetic operations. The expression is x^2 + 5x - 2. Substitute x with 11: (11)^2 + 5(11) - 2 Now, calculate the values: 11^2 = 121 5(11) = 55 So, the expression becomes: 121 + 55 - 2 Now add and subtract the numbers: 121 + 55 = 176 176 - 2 = 174 Therefore, the value of the expression when x = 11 is 174.
Solving Expressions Using Distributive Property
To solve the expression given in the image, you'll need to apply the distributive property, which states that \( a(b + c) = ab + ac \). So for the expression \( 4(8n + 2) \), use the distributive property as follows: \( 4 \times 8n + 4 \times 2 \) Now multiply the numbers: \( 32n + 8 \) The simplified form of the expression is \( 32n + 8 \).
Solving for Expression with Given Values
To solve the question, we need to find which expression equals \( \frac{10}{3} \) when \( x = \frac{2}{3} \). Let's start by evaluating each expression with \( x = \frac{2}{3} \): A) \( 4x - 2 \) \[ 4\left(\frac{2}{3}\right) - 2 = \frac{8}{3} - 2 = \frac{8}{3} - \frac{6}{3} = \frac{2}{3} \] B) \( 2 - 8x \) \[ 2 - 8\left(\frac{2}{3}\right) = 2 - \frac{16}{3} = \frac{6}{3} - \frac{16}{3} = -\frac{10}{3} \] C) \( 8x - 6 \) \[ 8\left(\frac{2}{3}\right) - 6 = \frac{16}{3} - 6 = \frac{16}{3} - \frac{18}{3} = -\frac{2}{3} \] D) \( 5x \) \[ 5\left(\frac{2}{3}\right) = \frac{10}{3} \] Option D, \( 5x \), gives us the correct value of \( \frac{10}{3} \) when \( x = \frac{2}{3} \). Therefore, the correct answer is D) \( 5x = \frac{10}{3} \).
Solving or Simplifying Expressions
The expression in the image is r * 3a^2 - 3. To proceed with solving or simplifying this expression, we need additional information or a specific question regarding what to do with it. As it stands, this expression cannot be simplified any further without additional context or instructions. If this is part of an equation, or if we are asked to evaluate it for given values of r and a, then more can be done. Otherwise, the expression remains as it is.
Solving an Expression Using Order of Operations
To solve the equation in the image, you should follow the order of operations, commonly abbreviated as PEMDAS/BODMAS which stands for Parentheses/Brackets, Exponents/Orders, Multiplication-Division (left to right), Addition-Subtraction (left to right). The equation provided is: 8 ÷ 2(2 + 2) First, we solve the expression inside the parentheses: 2 + 2 = 4 Next, we apply multiplication or division from left to right: 8 ÷ 2 * 4 Now, we divide first since division and multiplication have the same precedence and we work from left to right: 8 ÷ 2 = 4 Then we multiply: 4 * 4 = 16 So the answer is: 16
Solving a Mathematical Expression with Order of Operations
This mathematical expression should be solved following the order of operations, which is often abbreviated as PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). The expression given is: 8 ÷ 2(2 + 2) First, solve the operation within parentheses (2 + 2), which equals 4. 8 ÷ 2 * 4 Then perform the division and multiplication from left to right. 8 ÷ 2 = 4 4 * 4 = 16 The result is 16.
Solving a Mathematical Expression
The expression in the image is: 6 ÷ 2(1 + 2) To solve this expression, we need to follow the order of operations, often remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Following this order: 1. Parentheses first: Calculate the expression inside the parentheses (1 + 2) = 3. Now the expression is: 6 ÷ 2 * 3 2. Multiplication and Division are of equal precedence and are performed from left to right. So we'll divide 6 by 2 and then multiply by 3: First do the division: 6 ÷ 2 = 3 Then do the multiplication: 3 * 3 = 9 So, the correct answer to the expression 6 ÷ 2(1 + 2) = 9.
Solving Mathematical Expression using PEMDAS
To solve the expression \(6 \div 2(1 + 2)\), follow the order of operations, which is commonly known by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right). Let's solve it step by step: 1. **Parentheses**: First calculate the expression inside the parentheses. \(1 + 2 = 3\) 2. **Multiplication/Division**: Since there is a division and multiplication by 2, and no explicit operation between the 2 and the parentheses, we treat them as a multiplication. In many interpretations according to standard mathematical conventions, the expression next to the parentheses indicates multiplication, particularly when there is an implied multiplication as in this case. So, we rewrite the expression as: \(6 \div 2 \times 3\) Since multiplication and division are of the same precedence, we work from left to right. First, divide 6 by 2: \(6 \div 2 = 3\) Then multiply the result by 3: \(3 \times 3 = 9\) Therefore, the final answer to the expression is \(9\).
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