Example Question - mathematical simplification
Here are examples of questions we've helped users solve.
Simplifying Cubic Roots of Variables
Para resolver la expresión matemática dada en la imagen, primeramente simplifiquemos la raíz cúbica de los números y variables. La raíz cúbica de 64 es 4, porque \(4^3 = 4 \times 4 \times 4 = 64.\) La raíz cúbica de \(x^{16}\) es \(x^{16/3}\) porque al elevarla al cubo obtenemos \( (x^{16/3})^3 = x^{16}.\) La raíz cúbica de \(y^{13}\) es \(y^{13/3}\) porque al elevarla al cubo obtenemos \( (y^{13/3})^3 = y^{13}.\) Por lo tanto, la expresión completa bajo la raíz cúbica se simplifica como: \( \frac{3}{2} \times 4 \times x^{16/3} \times y^{13/3}.\) Multiplicamos ahora el coeficiente fuera de la raíz, 3/2, por 4 (que es la raíz cúbica de 64): \( \frac{3}{2} \times 4 = \frac{3 \times 4}{2} = \frac{12}{2} = 6. \) Entonces, la expresión simplificada es: \( 6 \times x^{16/3} \times y^{13/3} \). No podemos simplificar las variables más porque no tenemos más información sobre los valores de x o y. Por lo tanto, llegamos al resultado final con la expresión simplificada: \( 6x^{16/3}y^{13/3}. \)
Simplifying an Algebraic Expression
The image shows an algebraic expression that needs to be simplified: \[ \frac{1 - \sqrt{2}}{\sqrt{5} - \sqrt{3}} + \frac{1 + \sqrt{2}}{\sqrt{5} + \sqrt{3}} \] To simplify this expression, we can combine the two fractions into a single fraction by finding a common denominator, which is \((\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3})\). First, we recognize that \((\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3})\) equals \(\sqrt{5}^2 - \sqrt{3}^2\), which simplifies to \(5 - 3\), or \(2\). Now, let's work out the combined numerator: For denominator \(\sqrt{5} - \sqrt{3}\), the corresponding numerator is \((1 - \sqrt{2})(\sqrt{5} + \sqrt{3})\). For denominator \(\sqrt{5} + \sqrt{3}\), the corresponding numerator is \((1 + \sqrt{2})(\sqrt{5} - \sqrt{3})\). Let's find each product for the numerators separately: - \((1 - \sqrt{2})(\sqrt{5} + \sqrt{3}) = \sqrt{5} + \sqrt{3} - \sqrt{10} - \sqrt{6}\) - \((1 + \sqrt{2})(\sqrt{5} - \sqrt{3}) = \sqrt{5} - \sqrt{3} + \sqrt{10} - \sqrt{6}\) Next, combine these to get the new numerator: \[ (\sqrt{5} + \sqrt{3} - \sqrt{10} - \sqrt{6}) + (\sqrt{5} - \sqrt{3} + \sqrt{10} - \sqrt{6}) \] Simplify by cancelling out the opposite terms \( \sqrt{3} \) and \( -\sqrt{3} \), and by combining like terms: \[ 2\sqrt{5} - 2\sqrt{6} \] With the denominator found to be \(2\), we now have: \[ \frac{2\sqrt{5} - 2\sqrt{6}}{2} \] Divide both terms in the numerator by the denominator to simplify further: \[ \sqrt{5} - \sqrt{6} \] Thus, the simplified expression is: \[ \sqrt{5} - \sqrt{6} \]
Simplified Expression with Square Roots and Cubes
The expression given in the image is: \[ \frac{2y\sqrt{x^3} + y^3 - 2x \times 2y}{2\sqrt{x^3} + y^3} \div \sqrt{x^3 + y^3} \] Let's simplify it step by step. First, we can simplify the numerator by combining like terms: \[2y\sqrt{x^3} + y^3 - 2x \times 2y = 2y\sqrt{x^3} + y^3 - 4xy\] Now, we recognize that \(\sqrt{x^3}\) is \(x^{3/2}\). This simplifies the expression to: \[ \frac{2yx^{3/2} + y^3 - 4xy}{2x^{3/2} + y^3} \div \sqrt{x^3 + y^3} \] Next, let's simplify the divisor. \(\sqrt{x^3 + y^3}\) is the root of the sum of cubes, and it is not factorable in the same simple way as the sum of squares. So, we leave it as is. Now we can rewrite the division as multiplication by the reciprocal: \[ \frac{2yx^{3/2} + y^3 - 4xy}{2x^{3/2} + y^3} \times \frac{1}{\sqrt{x^3 + y^3}} \] Now we multiply the fractions: \[ \frac{(2yx^{3/2} + y^3 - 4xy) \times 1}{(2x^{3/2} + y^3) \times \sqrt{x^3 + y^3}} \] Since there are no like terms and no obvious factorizations to simplify further, and we're not given values for \(x\) or \(y\), this seems to be as simplified as we can get without further instructions or context. Therefore, the simplified form of the given expression is: \[ \frac{2yx^{3/2} + y^3 - 4xy}{(2x^{3/2} + y^3) \sqrt{x^3 + y^3}} \]
Mathematical Expressions Evaluation and Simplification
The image contains two different mathematical expressions that need to be evaluated or simplified: 1. Evaluate \(\cos^{-1}(\frac{1}{2}+\frac{\sqrt{3}}{2})\) 2. Simplify \(\frac{4^{-2} \times 2x - 2}{4x - 3 \times 0.5}\) Starting with the first one: 1. Evaluate \(\cos^{-1}(\frac{1}{2}+\frac{\sqrt{3}}{2})\) The expression inside the cosine inverse function is not possible since the cosine values are constrained between -1 and 1, and \(\frac{1}{2} + \frac{\sqrt{3}}{2}\) exceeds this range. Hence, the expression given is incorrect as it stands. If this is a mistake and you meant \(\cos^{-1}(\frac{1}{2})\) or \(\cos^{-1}(\frac{\sqrt{3}}{2})\), the results would be \(60^{\circ}\) or \(30^{\circ}\) respectively because cos(60°) = 1/2 and cos(30°) = √3/2. 2. Simplify \(\frac{4^{-2} \times 2x - 2}{4x - 3 \times 0.5}\) First simplify \(4^{-2}\). This is the same as \(1/4^{2}\) which equals \(1/16\). The expression now is: \(\frac{1/16 \times 2x - 2}{4x - 1.5}\) This simplifies to: \(\frac{(1/16) \times 2x - 2}{4x - 1.5}\) \(\frac{(1/8)x - 2}{4x - 1.5}\) There aren't any common factors between the numerator and the denominator, so this expression is simplified as much as possible given the current form. If additional context is provided or if there are any restrictions on the values of \(x\), further simplification may be possible.
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com