Example Question - square root expression
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Simplification of a Square Root Expression
Bien sûr, procédons à la simplification de l'expression donnée par étapes. Le nombre \( A \) est défini comme suit: \[ A = \frac{\sqrt{32} \times \sqrt{27} \times \sqrt{108}}{\sqrt{6}} \] Avant de simplifier, rappelons que lorsque l’on multiplie des racines carrées, on peut combiner les termes sous une seule racine. C’est-à-dire que \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \). 1. Multiplions d’abord les nombres sous les racines carrées: \[ A = \frac{\sqrt{32 \times 27 \times 108}}{\sqrt{6}} \] 2. Ensuite, calculez le produit de 32, 27, et 108: \[ 32 \times 27 \times 108 = 93312 \] 3. Remplacez ce produit dans la racine carrée: \[ A = \frac{\sqrt{93312}}{\sqrt{6}} \] 4. Maintenant, décomposons 93312 et 6 en facteurs premiers pour simplifier la racine carrée: \[ 93312 = 2^{6} \times 3^{3} \] \[ 6 = 2^1 \times 3^1 \] 5. Réécrivons l'expression en utilisant ces facteurs premiers: \[ A = \frac{\sqrt{2^{6} \times 3^{3}}}{\sqrt{2 \times 3}} \] 6. Nous pouvons maintenant simplifier la fraction en annulant les racines communes: \[ A = \frac{2^{6/2} \times 3^{3/2}}{2^{1/2} \times 3^{1/2}} \] 7. Simplifions les exposants: \[ A = 2^{3} \times 3^{\frac{3}{2}} \div 2^{\frac{1}{2}} \times 3^{\frac{1}{2}} \] \[ A = 2^{3 - \frac{1}{2}} \times 3^{\frac{3}{2} - \frac{1}{2}} \] \[ A = 2^{\frac{5}{2}} \times 3^1 \] 8. Pour finir, calculez les valeurs finales et convertissez les racines carrées en leur forme exponentielle: \[ A = 2^2 \times 2^{\frac{1}{2}} \times 3 \] \[ A = 4 \times \sqrt{2} \times 3 \] 9. Enfin, multiplions les nombres entiers: \[ A = 12\sqrt{2} \] La valeur simplifiée de \( A \) est donc \( 12\sqrt{2} \).
Solving Square Root Expression Division
The expression in question is the square root of B divided by the square root of J. Mathematically, we can express this as: √(B/J) Now, according to the properties of square roots, the square root of a quotient is equal to the quotient of the square roots of the numerator and the denominator. So one can rewrite the expression as: √B/√J Hence, the correct answer is C: √B/√J.
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.
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