Example Question - expression evaluation
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Evaluating an Algebraic Expression Involving Square Roots
Given the equation \(2\sqrt{2} + \sqrt{125} - \sqrt{45} + 4 = a + b\sqrt{c}\), First, simplify the square roots that are not in simplest form: <p>\(\sqrt{125} = \sqrt{25 \cdot 5} = 5\sqrt{5}\),</p> <p>\(\sqrt{45} = \sqrt{9 \cdot 5} = 3\sqrt{5}\).</p> Now, substitute these simplified forms back into the equation: <p>\(2\sqrt{2} + 5\sqrt{5} - 3\sqrt{5} + 4 = a + b\sqrt{c}\).</p> Combine like terms: <p>\(2\sqrt{2} + (5\sqrt{5} - 3\sqrt{5}) + 4 = a + b\sqrt{c}\),</p> <p>\(2\sqrt{2} + 2\sqrt{5} + 4 = a + b\sqrt{c}\).</p> Now, match the terms on both sides of the equation: <p>\(a = 4\),</p> <p>\(b = 2\), \(c = 2\),</p> <p>\(b = 2\), \(c = 5\).</p> Since the square root terms must match (they are like terms), we can see that there are two terms with a square root on the left side, \(\sqrt{2}\) and \(\sqrt{5}\). These two terms must match the single square root term on right side, \(b\sqrt{c}\). This is only consistent if \(c\) is a product of the radicands \(2\) and \(5\). So this suggests that \(b\) should be the sum of the coefficients of the square root terms on the left, while \(c\) should be the product of the radicands: <p>\(b=2+2=4\),</p> <p>\(c=2 \cdot 5 = 10\).</p> Hence, the equation can be written as: <p>\(4 + 4\sqrt{10} = a + b\sqrt{c}\).</p> Now that \(a\), \(b\), and \(c\) have been found, evaluate \(2a - b\): <p>\(2a - b = 2(4) - 4 = 8 - 4 = 4\).</p> The value of \(2a - b\) is \(4\).
Algebraic Evaluation of an Expression with Exponents
Given the equation \(2^x = 3^y = 6^z\), we want to find the value of \(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\). <p>\(6^z = (2 \cdot 3)^z = 2^z \cdot 3^z\)</p> <p>Since \(2^x = 3^y = 6^z\), we can say \(2^x = 2^z \cdot 3^z\).</p> <p>Therefore, \(x = z \cdot (\log_2{2} + \log_2{3}) = z + z \cdot \log_2{3}\).</p> <p>Similarly, \(3^y = 2^z \cdot 3^z\) implies \(y = z \cdot (\log_3{2} + \log_3{3}) = z \cdot \log_3{2} + z\).</p> <p>We find \(\log_2{3}\) and \(\log_3{2}\) by changing the base:</p> <p>\(\log_2{3} = \frac{1}{\log_3{2}}\)</p> <p>\(x = z + z \cdot \log_2{3} = z + \frac{z}{\log_3{2}}\)</p> <p>\(y = z \cdot \log_3{2} + z\)</p> <p>We then calculate the sum:</p> <p>\(\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{1}{z + \frac{z}{\log_3{2}}} + \frac{1}{z \cdot \log_3{2} + z} + \frac{1}{z}\)</p> <p>By finding a common denominator, we have:</p> <p>\(\frac{\log_3{2}(\log_3{2} + 1) + 1 + \log_3{2}(\log_3{2} + 1)}{z(\log_3{2} + 1)}\)</p> <p>This simplifies to:</p> <p>\(\frac{2\log_3{2}(\log_3{2} + 1) + 1}{z(\log_3{2} + 1)}\)</p> <p>Since \(\log_3{2}(\log_3{2} + 1) = 1\), we get:</p> <p>\(\frac{2 \cdot 1 + 1}{z(\log_3{2} + 1)} = \frac{3}{z(\log_3{2} + 1)}\)</p> <p>Now, \(y = z \cdot \log_3{2} + z\) gives us \(y = z(\log_3{2} + 1)\), hence substituting \(y\) into the denominator yields:</p> <p>\(\frac{3}{y}\)</p> <p>Since \(2^x=3^y\), it follows that \(x=y\), thus \(\frac{3}{y} = \frac{3}{x}\).</p> <p>Finally, \(2^x=3^y\) implies \(1 = \frac{3}{x}\), so the value of the sum \(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\) is 3.</p>
Evaluate the Expression with Given Variable Values
Dado que \text{a} = 3, \text{b} = 4 \text{ y c} = 5, evaluamos la expresión 3\text{a} - 5\text{b} + 2\text{c} sustituyendo los valores correspondientes: <p>3\text{a} - 5\text{b} + 2\text{c} = 3(3) - 5(4) + 2(5)</p> <p>3\text{a} - 5\text{b} + 2\text{c} = 9 - 20 + 10</p> <p>3\text{a} - 5\text{b} + 2\text{c} = -11 + 10</p> <p>3\text{a} - 5\text{b} + 2\text{c} = -1</p> Por lo tanto, la solución es \text{-1}.
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.
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 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.
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