Example Question - square root
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Solving an arithmetic problem with basic operations and square root
<p>\(\frac{140}{2} + \sqrt{4 \cdot 3 + 2} \cdot 4 - (25-5-3)^2\)</p> <p>\(= 70 + \sqrt{12 + 2} \cdot 4 - 17^2\)</p> <p>\(= 70 + \sqrt{14} \cdot 4 - 289\)</p> <p>\(= 70 + 2 \sqrt{14} \cdot 4 - 289\)</p> <p>\(= 70 + 8 \sqrt{14} - 289\)</p> <p>\(= 70 + 8 \cdot 3.74 - 289\), as \(\sqrt{14} \approx 3.74\)</p> <p>\(= 70 + 29.92 - 289\)</p> <p>\(= 99.92 - 289\)</p> <p>\(= -189.08\)</p> <p>Therefore, the result is approximately \(-189.08\).</p>
Solve the System of Equations with Square Root and Quadratic Equations
<p>Определим, что \( \sqrt{x - x^2} = 4 - x y \). Возводим обе стороны в квадрат:</p> <p>\( x - x^2 = (4 - xy)^2 \)</p> <p>\( x - x^2 = 16 - 8xy + x^2y^2 \)</p> <p>Получим выражение для \( y^2 \) из второго уравнения системы:</p> <p>\( y^2 - 4xy + 4 = 0 \)</p> <p>\( y^2 = 4xy - 4 \)</p> <p>Подставим это выражение для \( y^2 \) в предыдущее уравнение:</p> <p>\( x - x^2 = 16 - 8xy + x(4xy - 4) \)</p> <p>\( x - x^2 = 16 - 8xy + 4x^2y - 4x \)</p> <p>Теперь упростим и решим получившееся уравнение относительно \( x \):</p> <p>\( x - x^2 = 16 - 8xy + 4x^2y - 4x \)</p> <p>\( 0 = 16 - 8xy + 4x^2y - 4x + x^2 - x \)</p> <p>\( 0 = 16 - 4x(2y - 1) + x^2(4y - 1) \)</p> <p>Далее, учитывая, что система имеет несколько решений, можно проанализировать и использовать подходы для решения квадратных уравнений, чтобы найти допустимые значения \( x \) и \( y \). В конечном итоге, необходимо решить систему уравнений с учетом возможных ограничений, введенных квадратным корнем и квадратными уравнениями.</p>
Calculating the Square Root of a Six-Digit Number
<p>Para resolver la raíz cuadrada del número 59439 mediante el método de división larga, seguimos los siguientes pasos:</p> <p>Paso 1: Agrupamos los dígitos en pares desde el punto decimal hacia la izquierda y hacia la derecha. En este caso es $\sqrt{59\ 439}$.</p> <p>Paso 2: Encontramos el mayor número cuyo cuadrado sea menor o igual a 59. Este número es 7, ya que $7^2=49$.</p> <p>Escribimos 7 arriba y restamos $49$ de $59$, lo cual nos da $10$.</p> <p>Paso 3: Bajamos el siguiente par de dígitos (43) y lo añadimos al residuo, dando $1043$.</p> <p>Paso 4: El divisor ahora es $7 \times 2 = 14$. Buscamos un dígito (llamémoslo "D") tal que $(140+D) \cdot D$ sea menor o igual a $1043$.</p> <p>Para $D=3$, $(140+3) \cdot 3=429$. Esto es menor que $1043$, y si subimos a $D=4$, tenemos $(140+4) \cdot 4=576$, que también es menor a $1043$. Probamos con $D=5$ y obtenemos $(140+5) \cdot 5=725$, que aún es menor a $1043$, pero con $D=6$ obtenemos $(140+6) \cdot 6=876$, y este es el valor máximo de D que es menor a $1043$.</p> <p>Escribimos 6 al lado de 7 arriba y restamos $876$ de $1043$, quedando $167$.</p> <p>Paso 5: Bajamos el siguiente par de dígitos (39) y lo añadimos al residuo, dando $16739$.</p> <p>Paso 6: El nuevo divisor será $76 \times 2 = 152$. Ahora buscamos un dígito (llamémoslo "E") tal que $(1520+E) \cdot E$ sea menor o igual a $16739$.</p> <p>Para $E=1$, $(1520+1)\cdot 1=1521$. Al subir a $E=2$, tenemos $(1520+2) \cdot 2=3044$, y así sucesivamente, hasta que encontramos que para $E=9$, $(1520+9) \cdot 9=13761$ es el valor más alto de E menor a $16739$.</p> <p>Finalemente, escribimos 9 al lado de 76 arriba y restamos $13761$ de $16739$, obteniendo un residuo de $2978$.</p> <p>Por lo tanto, los decimales hasta este punto del número son $769$.</p> <p>El procedimiento puede continuar para obtener más decimales si es necesario, pero con los dígitos dados en la pregunta, la raíz cuadrada aproximada de $59439$ es $769$.</p>
Square Root Calculation
<p>Para resolver la raíz cuadrada de 5439, podemos intentar factorizar el número hasta que se encuentre un cuadrado perfecto o utilizar una calculadora para una estimación. Sin embargo, para este ejemplo, utilizaremos la descomposición en factores primos para simplificar la raíz cuadrada tanto como sea posible.</p> <p>Descomponiendo 5439 en factores primos obtenemos:</p> <p>5439 = 3 * 17 * 107</p> <p>Como ninguno de los factores primos aparece dos veces, no hay cuadrados perfectos y no se puede simplificar la raíz cuadrada en términos de enteros. Por lo tanto, la solución final es simplemente la raíz cuadrada del número original:</p> <p>\(\sqrt{5439} \approx 73.752\)</p>
Simplify Radical and Fractional Expression
<p>\( \sqrt{8} + \frac{4}{\sqrt{2}} \)</p> <p>\( = \sqrt{4 \cdot 2} + \frac{4}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} \)</p> <p>\( = 2\sqrt{2} + \frac{4\sqrt{2}}{2} \)</p> <p>\( = 2\sqrt{2} + 2\sqrt{2} \)</p> <p>\( = 4\sqrt{2} \)</p>
Simplifying a Radical Expression
<p>\sqrt[3]{64} - (-\sqrt[3]{-64})</p> <p>= 4 - (-(-4))</p> <p>= 4 + 4</p> <p>= 8</p>
Finding Equivalent Expression for the Square Root
To find another name for the square root expression given, we should look for an expression that is mathematically equivalent. The original expression is: \( \sqrt{\frac{B}{J}} \) We're looking for an alternative expression that has the same value. Multiplying or dividing both the numerator and the denominator by the same nonzero number will keep the fraction equivalent. When we take the square root of a fraction, we can take the square root of the numerator and the denominator separately. Therefore, the expression can also be written as: \( \frac{\sqrt{B}}{\sqrt{J}} \) This corresponds to option C in the image provided: C) \( \frac{\sqrt{B}}{\sqrt{J}} \)
Simplifying a Mathematical Expression Involving Roots
Claro, vamos a resolver la expresión paso a paso. Primero, abordemos la parte izquierda de la ecuación: \[ \sqrt[3]{\frac{3^4}{2^2}} \cdot \sqrt[3]{\sqrt[3]{64}} = \sqrt[3]{\frac{81}{4}} \cdot \sqrt[3]{\sqrt[3]{4^3}} \] Como \(3^4 = 81\) y \(2^2 = 4\), la primera raíz cúbica se simplifica a \(\sqrt[3]{\frac{81}{4}}\). Además, como \(64 = 4^3\), la segunda expresión se simplifica a \(\sqrt[3]{4}\): \[ \sqrt[3]{\frac{81}{4}} \cdot \sqrt[3]{4} = \frac{\sqrt[3]{81}}{\sqrt[3]{4}} \cdot \sqrt[3]{4} = \sqrt[3]{81} = 3^{4/3} = 3^{1\frac{1}{3}} = 3 \cdot 3^{1/3} \] Ahora, abordemos la parte derecha de la ecuación: \[ \sqrt{2^2 + 6} = \sqrt{4 + 6} = \sqrt{10} \] Ahora, sumamos las dos partes para obtener la expresión completa: \[ 3 \cdot 3^{1/3} + \sqrt{10} \] Y esta es la forma simplificada de la expresión original. No se puede simplificar más sin conocer el valor decimal de las raíces.
Finding the Integer Closest to the Cube Root of the Square Root of 2
The expression in the image is asking for the cube root of the square root of 2, written as ∛√2. This can also be written as 2^(1/3 * 1/2), which simplifies to 2^(1/6). Now, to find the integer closest to 2^(1/6), we can try to estimate this value. Since 2^(1/6) is the sixth root of 2, we're looking for a number which, when raised to the power of 6, is closest to 2. Starting with 1, we see that 1^6 = 1, which is far from 2. Next, let's take 2^6, which is 64; this is much larger than 2. Therefore, the integer closest to 2^(1/6) must be 1, since any number greater than 1 raised to the power of 6 will be greater than 2. So, the integer closest to ∛√2 is 1.
Solving Inequality with Square Root
The given image depicts an inequality with a square root in the center: two blank boxes with less than signs around the square root of 104, written as "< √104 <". To solve this, we need to find two consecutive whole numbers between which the square root of 104 falls. Since we know that \(10^2 = 100\) and \(11^2 = 121\), the square root of 104 must be greater than 10 but less than 11 because 104 falls between 100 and 121. Therefore, the correct numbers to place in the boxes are 10 and 11. The completed inequality should read "10 < √104 < 11".
Solving Inequality with Square Root
The image shows an inequality with two blank boxes and a square root sign in between. The inequality is structured as a number less than the square root of 86, which is then less than another number. To solve for the inequality, you first need to calculate the square root of 86 to determine what numbers the blanks could represent: √86 ≈ 9.27 (approximately, as the square root of 86 cannot be simplified to a simple integer) The numbers in the blank boxes should be close to but not equal to the square root of 86, with the left number being less than 9.27 and the right number being greater than 9.27. One potential set of numbers that would satisfy these conditions could be: Left box: 9 Right box: 10 Since 9 < √86 < 10 holds true with √86 being approximately 9.27.
Understanding Complex Numbers and Inequalities
The expression in the image includes the square root of a negative number: \(-\sqrt{86}\). In the realm of real numbers, you cannot take the square root of a negative number, as square roots are defined only for non-negative numbers. However, in the context of complex numbers, the square root of a negative number is a multiple of the imaginary unit \(i\), where \(i\) is defined as \(i^2 = -1\). Thus, \(-\sqrt{86}\) in terms of complex numbers is \(-i\sqrt{86}\). This number is purely imaginary, and there is no real number less than or greater than it because comparisons of greater or less than do not apply to imaginary numbers. Therefore, the blank boxes on both sides of \(-\sqrt{86}\) cannot be filled with real numbers to create a true statement about inequalities. Filling these blanks would require a context where complex numbers are included, and even then, the notion of "greater than" or "less than" isn't meaningful in the same way as it is with real numbers.
Completing a Mathematical Inequality with Square Root
The image shows an incomplete mathematical inequality with a square root. We see \( \sqrt{68} \) and two blank spaces, one to the left and one to the right of this square root, waiting to be filled with appropriate numbers to complete the inequality. To find numbers that fit this inequality, we first need to estimate the value of \( \sqrt{68} \). Since \( 8^2 = 64 \) and \( 9^2 = 81 \), we know that \( \sqrt{68} \) falls between 8 and 9. More precisely, since 68 is closer to 64 than it is to 81, \( \sqrt{68} \) will be closer to 8 than to 9. Therefore, to satisfy the inequality, we can choose a number less than 8 for the blank on the left (for example, 7) and a number between 8 and 9 for the blank on the right (such as 8.5). So, one possible way to fill the blanks would be: \[ 7 < \sqrt{68} < 8.5 \] Note that these are just example numbers. Any number less than 8 could fit into the first blank, and any number between 8 and 9 (but not including 9) could fit into the second blank.
Solving an Equation with Square Root
Bài toán trong hình đặt ra là: Giải phương trình: \(x^2 - 3x + 2 - \sqrt{x - 1} = x^2 - 3x + 2\) Để giải phương trình này, trước hết ta xác định điều kiện xác định của phương trình: \(\sqrt{x - 1}\) tồn tại \( \rightarrow x - 1 \geq 0 \rightarrow x \geq 1\). Do hai vế của phương trình đều có dạng \(x^2 - 3x + 2\), nên ta có thể tách riêng phần chứa căn thức để giải: \(x^2 - 3x + 2 - (x^2 - 3x + 2) + \sqrt{x - 1} = 0\) \(\sqrt{x - 1} = 0\) Bây giờ, ta tìm giá trị của \(x\) để phương trình trên được thoả mãn: \(\sqrt{x - 1} = 0 \rightarrow x - 1 = 0 \rightarrow x = 1\) Kiểm tra điều kiện, ta thấy \(x = 1\) thoả mãn điều kiện xác định, nên đây là nghiệm của phương trình. Vậy phương trình có một nghiệm duy nhất là \(x = 1\).
Solving Equations Involving Imaginary Numbers
To solve the equation \( x^2 = -169 \), you'll need to take the square root of both sides. However, because the right side of the equation is negative, it means that you will be dealing with imaginary numbers, since the square root of a negative number is not defined within the real numbers. Here are the steps: \[ x^2 = -169 \] Take the square root of both sides: \[ \sqrt{x^2} = \sqrt{-169} \] Since \( \sqrt{x^2} = x \) and \( \sqrt{-169} = \sqrt{-1} \cdot \sqrt{169} \), and \( \sqrt{-1} \) is defined as the imaginary unit \( i \), you'll have: \[ x = \pm 13i \] The \( \pm \) symbol indicates that there are two solutions for \( x \), one positive and one negative. So the solutions to the equation are \( x = 13i \) and \( x = -13i \).
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