Example Question - evaluation
Here are examples of questions we've helped users solve.
Mathematical Statements Evaluation
<p>Para la opción a:</p> <p>6 + 4 = 10 y 9 - 4 = 5;</p> <p>Ambas afirmaciones son verdaderas.</p> <p>Para la opción b:</p> <p>8/2 = 4 y 8 + 2 = 12;</p> <p>Ambas afirmaciones son verdaderas.</p> <p>Para la opción c:</p> <p>No se puede evaluar como verdadera o falsa ya que es una afirmación.</p> <p>Para la opción d:</p> <p>Si 3 * 7 = 21, entonces 9 - 7 = 2;</p> <p>Ambas afirmaciones son verdaderas.</p>
Calculating a Complex Expression
<p>Primero, evaluamos la expresión:</p> <p>E = 10^{32} \cdot 25^{8} \cdot 3^{-1} + \left(-16^{-4} \cdot 0.29^{5}\right)</p> <p>Calculemos cada parte por separado:</p> <p>1. Calculamos \(10^{32}\), \(25^{8}\) y \(3^{-1}\).</p> <p>2. Multiplicamos los resultados de \(10^{32} \cdot 25^{8} \cdot 3^{-1}\).</p> <p>3. Evaluamos \(-16^{-4}\) y \(0.29^{5}\).</p> <p>4. Multiplicamos los resultados de \(-16^{-4} \cdot 0.29^{5}\).</p> <p>5. Finalmente, sumamos los dos resultados para encontrar \(E\).</p>
Calculation of Expressions
<p>Para resolver la expresión dada:</p> <p> \( E = 100^{3^2} \cdot 25 \cdot 8^{3 - 1} + \left( \frac{1}{81} \right)^{-16} \cdot 0.29^3 \) </p> <p>Primero, cálculos intermedios:</p> <p>100 es \( 10^2 \), entonces \( 100^{3^2} = (10^2)^{9} = 10^{18} \)</p> <p>25 es \( 5^2 \), entonces \( 25 \) permanece igual.</p> <p>8 es \( 2^3 \), entonces \( 8^{3-1} = 8^2 = 64 \) o \( (2^3)^2 = 2^6 = 64 \).</p> <p>Ahora, \( E = 10^{18} \cdot 25 \cdot 64 \)</p> <p>Ahora, calcularemos \( ( \frac{1}{81} )^{-16} = 81^{16} \) y \( 0.29^3 \).</p> <p>Finalmente, sumamos ambos resultados para encontrar \( E \).</p>
Algebraic Expression Evaluation
<p>For the expressions given:</p> <p>1. \(4(y-2) = 4y - 8\)</p> <p>2. \(5f(2f+7) = 10f^2 + 35f\)</p> <p>3. \((5x-4)(2x+3) = 10x^2 + 15x - 8x - 12 = 10x^2 + 7x - 12\)</p>
Algebraic Expressions Evaluation
<p>To evaluate the expressions:</p> <p>1. \( 4(y-2) = 4y - 8 \)</p> <p>2. \( 5f(2f+7) = 10f^2 + 35f \)</p> <p>3. \( (5x-4)(2x+3) = 10x^2 + 15x - 8 \)</p>
Analysis of a Limit as x Approaches Infinity
<p>\[ \lim_{{x \to +\infty}} \frac{2}{x} = \frac{2}{+\infty} \]</p> <p>\[ = 0 \]</p>
Evaluating Numerical Expressions in Decimal and Fraction Form
<p>問1：絶対値が2以下の実数をすべて表現しよう。</p> <p>この問題の答えは、実数 x に対して、-2 \leq x \leq 2 です。</p> <p>\[-2 \leq x \leq 2\]</p> <p>問2：絶対値が2以上5未満の実数を表しましょう。</p> <p>この問題の答えは、実数 x に対して、-5 < x < -2 または 2 \leq x < 5 です。</p> <p>\[-5 < x < -2 または 2 \leq x < 5\]</p> <p>問3：次の２数の大小を、不等号を使って表しましょう。</p> <p>この問題の答えは以下の通りです。</p> <p>\[\frac{-1}{2} > -0.6\]</p> <p>\[-1 < -0.6\]</p> <p>従って、\[-1, -0.6\] の順に小さい方から大きい方へと並べることができます。</p>
Evaluating the Sum of Integers with Negative Signs
<p>La expresión matemática es la suma de dos números enteros negativos.</p> <p>Para resolver la suma de dos números negativos, simplemente sumamos los valores absolutos de los números y mantenemos el signo negativo:</p> <p>\[ (-7) + (-9) = -7 + -9 \]</p> <p>\[ = -(7 + 9) \]</p> <p>\[ = -(16) \]</p> <p>\[ = -16 \]</p> <p>Por lo tanto, el resultado de \( (-7) + (-9) \) es \(-16\).</p>
Solving Algebraic Expressions
10. z*y - 5 11. x/24 12. 6*y + 2
Solving Definite Integral of x
Para resolver la integral definida \(\int_{0}^{a} x dx\), primero encontraremos la integral indefinida de \(x\) con respecto a \(x\), y después evaluaremos esa integral entre los límites de 0 a \(a\). La integral indefinida de \(x\) es \(\frac{x^2}{2}+C\), donde \(C\) es la constante de integración. Dado que estamos haciendo una integral definida, la constante \(C\) se cancelará al evaluar en los límites, así que podemos ignorarla. Ahora evaluamos la integral indefinida entre los límites dados: \[ \left[\frac{x^2}{2}\right]_0^a = \frac{a^2}{2} - \frac{0^2}{2} = \frac{a^2}{2} - 0 = \frac{a^2}{2} \] Por lo tanto, la solución de la integral es \(\frac{a^2}{2}\). La respuesta correcta es la opción c) \(\frac{a^2}{2}\).
Evaluation of Sin and Square Root Expression
The expression you're trying to evaluate is "sin(π/2) √72". Let's break it down step by step. First, let's find the value of sin(π/2), which is the sine of 90 degrees. The sine of a 90-degree angle (π/2 radians) is equal to 1. Next, let's simplify the square root of 72. The square root of 72 can be broken down into √(36 * 2), which simplifies to 6√2, because √36 equals 6. Multiplying these two results together (since sin(π/2) = 1 doesn't really affect the multiplication): 1 * 6√2 = 6√2 So the final answer to the expression sin(π/2) √72 is 6√2.
Understanding Inverse Trigonometric Functions
The question in the image asks to evaluate the expression: \[ \cos(\cos^{-1}(-0.9)) \] By definition, the inverse cosine function, \(\cos^{-1}(x)\), returns the angle whose cosine is \(x\). Therefore, when we apply the cosine function to the result of an inverse cosine function, we get back the original value inside the inverse cosine. For the given problem: \[ \cos(\cos^{-1}(-0.9)) = -0.9 \] This is because the inverse cosine of \(-0.9\) gives us the angle whose cosine is \(-0.9\), and taking the cosine of that angle returns us back to the original value of \(-0.9\). So the correct answer is: -0.9
Simplified Evaluation of Complex Number Expression
The expression you're being asked to evaluate is: (1 + 3i)^8 + (1 - 3i)^8 This is an expression involving complex numbers. When raised to powers, complex numbers can sometimes simplify due to their periodic nature in the complex plane, but it is usually not feasible to compute high powers like 8 by hand without a significant amount of computation. In this case, there's a shortcut that can simplify the process using binomial expansion and the fact that i^2 = -1. To evaluate this expression, you would normally apply the binomial theorem, which states that: (a + b)^n = Σ[k=0 to n] (n choose k) * a^(n-k) * b^k Where "n choose k" is the binomial coefficient, calculated as: (n choose k) = n! / (k! * (n - k)!) However, for the powers of 8, this would involve a lot of terms and computation, namely: (1 + 3i)^8 = 1^8 + (8 choose 1)*1^7*(3i) + (8 choose 2)*1^6*(3i)^2 + ... + (3i)^8 (1 - 3i)^8 = 1^8 + (8 choose 1)*1^7*(-3i) + (8 choose 2)*1^6*(-3i)^2 + ... + (-3i)^8 You will notice that terms with odd powers of (3i) in (1 + 3i)^8 will cancel out with the corresponding terms in (1 - 3i)^8 due to the opposite signs. Only the even powers will remain, which are identical in both expressions because (-3i)^2n = (3i)^2n for all integers n, as both will be a real number. To simplify calculation, you can just compute the terms involving even powers of i from either (1 + 3i)^8 or (1 - 3i)^8 and then double the real part to get the final answer. However, for powers of 8 and without further tricks, this could be a tedious process and is typically done using a computer algebra system. For simplicity and practicality, let me assist you in solving this with such computational help: (1 + 3i)^8 + (1 - 3i)^8 simplifies to 2 * (1^8 + (8 choose 2)*1^6*(3i)^2 + (8 choose 4)*1^4*(3i)^4 + (8 choose 6)*1^2*(3i)^6 + (3i)^8). Each individual term can be computed, keeping in mind that i^2 = -1, i^4 = i^2 * i^2 = (-1)^2 = 1, i^6 = i^4 * i^2 = 1 * (-1) = -1, and i^8 = i^6 * i^2 = (-1) * (-1) = 1. Here's the simplification of the even-i-powered terms: (8 choose 2)(3i)^2 = 28 * 9 * (-1) = -252 (8 choose 4)(3i)^4 = 70 * 81 * 1 = 5670 (8 choose 6)(3i)^6 = 28 * 729 * (-1) = -20412 (3i)^8 = 6561 * 1 = 6561 Summing these and multiplying by 2 gives the real part: 2 * (1 - 252 + 5670 - 20412 + 6561) = 2 * (1 - 252 + 5670 - 20412 + 6561) = 2 * 51568 = 103136 So the final answer is: 103136
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