Example Question - simplify expression
Here are examples of questions we've helped users solve.
Calculating a Missing Number
First, simplify the expression inside the brackets: \( 5.05 + \frac{2}{5} = 5.05 + 0.4 = 5.45 \) Next, multiply the result by \(-3.5\): \( 5.45 \times (-3.5) = -19.075 \) Now, to find the missing number, divide by 2.4: \( \frac{-19.075}{2.4} = -7.9479166667 \) Rounded to two decimal places: \( -7.95 \)
Simplifying Expression with Radical Exponents
The image is rotated and not fully clear, but I will provide the steps for the equation that is visible: The problem looks like it involves simplifying the expression 2·3^(1/5) / (6^(2/5)·(3^(1/5))²) Step 1: Compute the exponent for 3^(1/5) when squared, which is (1/5) * 2 = 2/5. Step 2: Simplify the expression as such: 2·3^(1/5) / (6^(2/5)·3^(2/5)) Since 6 = 2·3, we can rewrite 6^(2/5) as (2·3)^(2/5), which is equivalent to 2^(2/5)·3^(2/5). 3: The expression is now: 2·3^(1/5) / (2^(2/5)·3^(2/5)) 4: We can now cancel out common terms: (2·3^(1/5)) / (2^(2/5)·3^(2/5)) = 3^(1/5) / 2^(2/5) 5: The solution will be the simplified form of 3^(1/5) / 2^(2/5). If you can provide a clearer image or clarify the exact expression, I can offer more precise help.
Simplifying Expression with Variables
На картинке представлено выражение: \(8p - 4g + 4p - g\) Чтобы решить это выражение, мы должны объединить подобные слагаемые. Подобные слагаемые — это те слагаемые, которые содержат одинаковые переменные в одинаковой степени. В данном случае, \(8p\) и \(4p\) являются подобными слагаемыми, так же как \(-4g\) и \(-g\). Сложим подобные слагаемые: \(8p + 4p = 12p\), \(-4g - g = -5g\). Теперь объединим их вместе: \(12p - 5g\). Это и есть упрощенный ответ выражения с картинки.
Simplified Expression Calculation
The expression given in the image is: \((x^2y^5 ÷ y^0)^2\) Firstly, let's simplify the expression inside the parentheses. Any nonzero number raised to the power of 0 is 1, which means \(y^0 = 1\). Therefore, our expression becomes: \((x^2y^5 ÷ 1)^2\) Since dividing by 1 does not change the value of the expression, we have: \((x^2y^5)^2\) Next, when you raise a power to another power, you multiply the exponents. Here's how to break it down: \((x^2)^2 * (y^5)^2\) Now calculate each term: \(x^2\) raised to the 2nd power is \(x^{2*2}\) which is \(x^4\), and \(y^5\) raised to the 2nd power is \(y^{5*2}\) which is \(y^{10}\). So after combining them, you get: \(x^4y^{10}\) This is the simplified form of the original expression.
Solving Math Expression and Finding Minimum Value
Dựa vào hình ảnh bạn cung cấp, chúng ta cần làm hai bước: a) Rút gọn biểu thức \( M \) b) Tìm giá trị nhỏ nhất của \( M \). Đầu tiên, chúng ta sẽ thực hiện rút gọn \( M \): \( M = \frac{1}{\sqrt{x} + \sqrt{x-1}} + \frac{1}{\sqrt{x} - \sqrt{x-1}} - \frac{4}{1-\sqrt{x}} \) Để rút gọn, ta nhân tử liên hợp cho mỗi phân thức: - Phân thức thứ nhất và thứ hai: \[ \frac{1}{\sqrt{x} + \sqrt{x-1}} \times \frac{\sqrt{x} - \sqrt{x-1}}{\sqrt{x} - \sqrt{x-1}} = \frac{\sqrt{x} - \sqrt{x-1}}{x - (x-1)} = \sqrt{x} - \sqrt{x-1} \] \[ \frac{1}{\sqrt{x} - \sqrt{x-1}} \times \frac{\sqrt{x} + \sqrt{x-1}}{\sqrt{x} + \sqrt{x-1}} = \frac{\sqrt{x} + \sqrt{x-1}}{x - (x-1)} = \sqrt{x} + \sqrt{x-1} \] - Phân thức thứ ba, nhân với tử liên hợp: \[ \frac{4}{1-\sqrt{x}} \times \frac{1+\sqrt{x}}{1+\sqrt{x}} = \frac{4(1+\sqrt{x})}{1 - x} \] Khi đó, \( M \) được rút gọn thành: \[ M = (\sqrt{x} - \sqrt{x-1}) + (\sqrt{x} + \sqrt{x-1}) - \frac{4(1+\sqrt{x})}{1 - x} \] Kết quả của hai phân thức đầu là \( 2\sqrt{x} \), ta được: \[ M = 2\sqrt{x} - \frac{4(1+\sqrt{x})}{1 - x} \] Giờ, ta đưa về cùng mẫu số: \[ M = \frac{2\sqrt{x}(1-x) - 4(1+\sqrt{x})}{1 - x} \] Đi tiếp, ta giản ước: \[ M = \frac{-2\sqrt{x} + 2x - 4 - 4\sqrt{x}}{1-x} \] Đặt \( \sqrt{x} = t \) (Với \( x \geq 0 \)), ta có \( x = t^2 \), và \( M \) trở thành: \[ M = \frac{-2t + 2t^2 - 4 - 4t}{1-t^2} \] \[ M = \frac{2t^2 - 6t - 4}{1-t^2} \] \[ M = \frac{2(t^2 - 3t - 2)}{- (t-1)(t+1)} \] \[ M = \frac{-2(t+1)(t-2)}{(t-1)(t+1)} \] \[ M = \frac{-2(t-2)}{(t-1)} \] \[ M = 2 - \frac{4}{t-1} \] Với \( t = \sqrt{x} \). b) Giờ đây, để tìm giá trị nhỏ nhất của \( M \), ta cần chú ý rằng mẫu số \( t - 1 \) phải luôn dương do \( x \) lớn hơn 1 nên \( t = \sqrt{x} > 1 \). Do đó, \( -4/(t-1) \) sẽ đạt giá trị lớn nhất khi \( t - 1 \) nhỏ nhất (nhưng vẫn dương), tức là khi \( t \) gần 1 nhất. Vì vậy, \( M \) đạt giá trị nhỏ nhất khi \( t \) càng gần với 1 từ bên phải. Tuy nhiên, \( M \) không có giá trị nhỏ nhất bởi vì khi \( t \) tiến về 1 thì \( -4/(t-1) \) sẽ tiến về âm vô cùng. Như vậy, theo phần a), ta có rút gọn được \( M \) và theo phần b), \( M \) không có giá trị nhỏ nhất.
Simplifying Algebraic Expression with Common Denominators
Para resolver la expresión dada en la imagen, primero simplificaremos los términos dentro de las fracciones y luego los combinaremos. La expresión es: \[ \frac{1}{10}m^3a - \frac{17}{60}m^2a^2 + \frac{3}{5}m^3 - \frac{7}{6}ma^3 - a^{4} \] Primero, necesitamos encontrar un común denominador para combinar los términos fraccionarios. El mínimo común denominador (m.c.m.) de 10, 60, 5, y 6 es 60. Ahora convertimos cada término a este denominador común: - \[ \frac{1}{10}m^3a = \frac{6}{60}m^3a \] - \[ \frac{17}{60}m^2a^2 \] ya tiene el común denominador, así que se queda igual. - \[ \frac{3}{5}m^3 = \frac{36}{60}m^3 \] - \[ \frac{7}{6}ma^3 = \frac{35}{60}ma^3 \] Ahora, la expresión con el común denominador es: \[ \frac{6}{60}m^3a - \frac{17}{60}m^2a^2 + \frac{36}{60}m^3 - \frac{35}{60}ma^3 - a^{4} \] Simplificamos combinando términos semejantes: \[ \frac{6m^3a - 17m^2a^2 + 36m^3 - 35ma^3}{60} - a^{4} \] Ya que todos los términos dentro de la fracción están divididos por 60, podemos escribir la expresión final como: \[ \frac{6m^3a + 36m^3 - 17m^2a^2 - 35ma^3}{60} - a^{4} \] Por último, para continuar simplificando, si es necesario, tendríamos que conocer el valor de las variables para resolver la expresión de manera numérica, o podemos dejar la expresión simbólica simplificada como está.
Rationalizing Denominators of Square Roots
To solve the given expression, we can use the technique of rationalizing the denominator. The expression given is: \[ \frac{\sqrt{2} + \sqrt{3}}{\sqrt{2} - \sqrt{3}} \] We will multiply both the numerator and denominator by the conjugate of the denominator to eliminate the square roots in the denominator. The conjugate of \(\sqrt{2} - \sqrt{3}\) is \(\sqrt{2} + \sqrt{3}\). Here's how it is done: \[ \frac{\sqrt{2} + \sqrt{3}}{\sqrt{2} - \sqrt{3}} \times \frac{\sqrt{2} + \sqrt{3}}{\sqrt{2} + \sqrt{3}} = \frac{(\sqrt{2} + \sqrt{3})^2}{(\sqrt{2})^2 - (\sqrt{3})^2} \] Expanding the numerator and simplifying the denominator: \[ \frac{2 + 2\sqrt{2}\sqrt{3} + 3}{2 - 3} = \frac{5 + 2\sqrt{6}}{-1} \] Simplifying further: \[ \frac{5 + 2\sqrt{6}}{-1} = -5 - 2\sqrt{6} \] The final simplified result is: \[ -5 - 2\sqrt{6} \]
Solving a Complex Mathematical Expression Step-by-Step
The image appears to be upside down; however, I can still read the content. It shows a mathematical problem which seems to be: \[ \frac{-9b^2 - 4 \times (12)(155)}{2 \times (12)} \] To solve this problem, follow these steps: 1. Multiply \(4 \times 12 \times 155\) to find the product in the numerator. 2. Once you've found the product, you will subtract it from \(-9b^2\) (still in the numerator). 3. Lastly, divide the result by \(2 \times 12\), which is the division in the denominator. Let's solve it step-by-step: 1. Compute \(4 \times 12 \times 155\) \[ 4 \times 12 = 48 \] \[ 48 \times 155 = 7440 \] 2. Subtract that product from \(-9b^2\) to calculate the new numerator. \[ -9b^2 - 7440 \] 3. Divide the new numerator by \(2 \times 12\), which is 24. \[ \frac{-9b^2 - 7440}{24} \] To simplify further, you can separate the fraction: \[ \frac{-9b^2}{24} - \frac{7440}{24} \] The terms may simplify depending on whether there are common factors. For the constant term: \[ \frac{7440}{24} = 310 \] Leaving us with: \[ \frac{-9b^2}{24} - 310 \] The fraction \(\frac{-9b^2}{24}\) can also be simplified by dividing both numerator and denominator by their greatest common divisor which is 3: \[ \frac{-9b^2}{24} = \frac{-3b^2}{8} \] So the simplified form of the expression would be: \[ \frac{-3b^2}{8} - 310 \] Please note that this expression may not simplify further without specific information about the variable \(b\).
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.
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.
Rationalizing the Denominator of a Complex Fraction
To express the given expression in the form \( q + r \sqrt{s} \), where \( q, r, \) and \( s \) are rational numbers, we need to rationalize the denominator. The given expression is: \[ \frac{2\sqrt{5} + 5\sqrt{2}}{2\sqrt{5} - 5\sqrt{2}} \] First, we can multiply the numerator and the denominator by the conjugate of the denominator to eliminate the square roots in the denominator. The conjugate of \(2\sqrt{5} - 5\sqrt{2}\) is \(2\sqrt{5} + 5\sqrt{2}\). Multiplying both the numerator and the denominator by the conjugate, we get: \[ \frac{(2\sqrt{5} + 5\sqrt{2})(2\sqrt{5} + 5\sqrt{2})}{(2\sqrt{5} - 5\sqrt{2})(2\sqrt{5} + 5\sqrt{2})} \] Now expand both the numerator and the denominator: The numerator, when multiplied out, becomes: \[ (2\sqrt{5} \cdot 2\sqrt{5}) + (2\sqrt{5} \cdot 5\sqrt{2}) + (5\sqrt{2} \cdot 2\sqrt{5}) + (5\sqrt{2} \cdot 5\sqrt{2}) \] \[ = 4 \cdot 5 + 2 \cdot 5 \cdot \sqrt{10} + 5 \cdot 2 \cdot \sqrt{10} + 25 \cdot 2 \] \[ = 20 + 10\sqrt{10} + 10\sqrt{10} + 50 \] \[ = 70 + 20\sqrt{10} \] The denominator, when multiplied out, becomes a difference of squares: \[ (2\sqrt{5})^2 - (5\sqrt{2})^2 \] \[ = 4 \cdot 5 - 25 \cdot 2 \] \[ = 20 - 50 \] \[ = -30 \] Combining the expanded numerator and the expanded denominator, we have: \[ \frac{70 + 20\sqrt{10}}{-30} \] Now, simplify this by dividing both terms in the numerator by the denominator: \[ \frac{70}{-30} + \frac{20\sqrt{10}}{-30} \] \[ = -\frac{7}{3} - \frac{2\sqrt{10}}{3} \] So the final answer, expressed in the form \( q + r \sqrt{s} \), is: \[ q = -\frac{7}{3}, \quad r = -\frac{2}{3}, \quad s = 10 \]
Expression Evaluation with Substitution
To evaluate the expression, simply substitute \( x = 20 \) into the expression and simplify: \[ \frac{5(x - 8)}{6} - 1 \] Plugging in \( x = 20 \): \[ \frac{5(20 - 8)}{6} - 1 \] \[ \frac{5(12)}{6} - 1 \] \[ \frac{60}{6} - 1 \] \[ 10 - 1 \] \[ 9 \] So, the value of the expression when \( x = 20 \) is 9.
Solving Cube Root of 81x^10y^8
The expression in the image is the cube root of \((81x^{10}y^8)\), which can be written as \((81x^{10}y^8)^{\frac{1}{3}}\). To simplify this expression, we take the cube root of each factor separately: 1. The cube root of 81, which is \(3^4\), is 3, because \(3^3 = 27\) and \(3^3 \times 3 = 81\). 2. The cube root of \(x^{10}\) can be simplified by dividing the exponent by 3. This gives us \(x^{\frac{10}{3}}\) or \(x^3 \times x^{\frac{1}{3}}\). 3. The cube root of \(y^8\) can be simplified by dividing the exponent by 3. This gives us \(y^{\frac{8}{3}}\) or \(y^2 \times y^{\frac{2}{3}}\). Putting it all together, you get: \(3x^3x^{\frac{1}{3}}y^2y^{\frac{2}{3}}\) You can also leave it as \(3x^{\frac{10}{3}}y^{\frac{8}{3}}\) if you prefer. Both forms are mathematically equivalent.
Simplified Cube Root of a Fraction
The expression in the image is: \[ \left(\frac{2 + 2\sqrt{13}}{6}\right)^3 \] To solve this expression, you need to raise the fraction within the parentheses to the power of 3. First, simplify the fraction by dividing both the numerator and the denominator by 2, then raise the result to the third power: Simplify the fraction: \[ \left(\frac{2}{6} + \frac{2\sqrt{13}}{6}\right)^3 = \left(\frac{1}{3} + \frac{\sqrt{13}}{3}\right)^3 \] Now compute the cube of the simplified expression. Let's denote \(a = \frac{1}{3}\) and \(b = \frac{\sqrt{13}}{3}\), then the expression becomes \((a + b)^3\). Using the binomial expansion: \[ (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \] Apply this formula: \[ \left(\frac{1}{3} + \frac{\sqrt{13}}{3}\right)^3 = \left(\frac{1}{3}\right)^3 + 3\left(\frac{1}{3}\right)^2\left(\frac{\sqrt{13}}{3}\right) + 3\left(\frac{1}{3}\right)\left(\frac{\sqrt{13}}{3}\right)^2 + \left(\frac{\sqrt{13}}{3}\right)^3 \] Now carry out the exponentiation and multiplication: \[ = \frac{1}{27} + 3 \cdot \frac{1}{9} \cdot \frac{\sqrt{13}}{3} + 3 \cdot \frac{1}{3} \cdot \frac{13}{9} + \frac{13\sqrt{13}}{27} \] Simplify the terms: \[ = \frac{1}{27} + \frac{\sqrt{13}}{9} + \frac{13}{9} + \frac{13\sqrt{13}}{27} \] Combine the terms: \[ = \frac{1 + 3\sqrt{13} + 39 + 13\sqrt{13}}{27} \] Combine like terms: \[ = \frac{40 + 16\sqrt{13}}{27} \] That's the simplified form of the cube of the original expression: \[ \left(\frac{2 + 2\sqrt{13}}{6}\right)^3 = \frac{40 + 16\sqrt{13}}{27} \]
Simplifying Algebraic Expressions with Exponents
To simplify the given expression, you'll need to use the laws of exponents. Let's simplify the numerator and the denominator separately first, and then we'll see if we can simplify further. The expression given is: \( \frac{10x^{5n+4} + 4.125x^{5n+2}}{3x^{5n+3} - 20x^{5n+1}} \) Numerator: \( 10x^{5n+4} + 4.125x^{5n+2} \) can be factored by taking out the common factor of \( x^{5n+2} \): \( = x^{5n+2} (10x^2 + 4.125) \) Denominator: \( 3x^{5n+3} - 20x^{5n+1} \) similarly can be factored by taking out the common factor of \( x^{5n+1} \): \( = x^{5n+1} (3x^2 - 20) \) Now the expression becomes: \( \frac{x^{5n+2} (10x^2 + 4.125)}{x^{5n+1} (3x^2 - 20)} \) Next, we'll divide the exponents: Since \( x^{5n+2} \) is divided by \( x^{5n+1} \), we subtract the exponents of like bases: \( x^{5n+2} ÷ x^{5n+1} = x^{(5n+2) - (5n+1)} = x^{1} = x \) Now we have: \( \frac{x(10x^2 + 4.125)}{3x^2 - 20} \) The expression is simplified to: \( \frac{10x^3 + 4.125x}{3x^2 - 20} \) This is as simple as the expression can get without further information about \( x \) or \( n \). If certain values are given for these variables, then numerical simplification could proceed. Otherwise, this is the simplified algebraic expression.
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com