Welcome to Westonci.ca, where your questions are met with accurate answers from a community of experts and enthusiasts. Explore a wealth of knowledge from professionals across various disciplines on our comprehensive Q&A platform. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
Sagot :
To solve the expression [tex]\(\frac{2}{3+\sqrt{2}+\sqrt{5}}\)[/tex], we need to rationalize the denominator. This involves eliminating the irrational numbers (square roots) from the denominator.
To achieve this, we will multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of a sum of square roots like [tex]\(3 + \sqrt{2} + \sqrt{5}\)[/tex] is usually taken by changing the signs of the square roots, so in this case, it will be [tex]\(3 - \sqrt{2} - \sqrt{5}\)[/tex].
Here are the steps in detail:
1. Identify the original expression:
[tex]\[ \frac{2}{3+\sqrt{2}+\sqrt{5}} \][/tex]
2. Multiply both the numerator and the denominator by the conjugate of the denominator:
[tex]\[ \frac{2}{3+\sqrt{2}+\sqrt{5}} \times \frac{3-\sqrt{2}-\sqrt{5}}{3-\sqrt{2}-\sqrt{5}} = \frac{2(3-\sqrt{2}-\sqrt{5})}{(3+\sqrt{2}+\sqrt{5})(3-\sqrt{2}-\sqrt{5})} \][/tex]
3. Expand the numerator:
[tex]\[ 2(3 - \sqrt{2} - \sqrt{5}) = 6 - 2\sqrt{2} - 2\sqrt{5} \][/tex]
4. Expand the denominator using the distributive property (a-b)(a+b) to handle the square roots:
[tex]\[ (3+\sqrt{2}+\sqrt{5})(3-\sqrt{2}-\sqrt{5}) \][/tex]
Let’s break it down using the distributive property:
[tex]\[ (3+\sqrt{2}+\sqrt{5})(3-\sqrt{2}-\sqrt{5}) \][/tex]
= [tex]\(3(3) + 3(-\sqrt{2}) + 3(-\sqrt{5}) + \sqrt{2}(3) + \sqrt{2}(-\sqrt{2}) + \sqrt{2}(-\sqrt{5}) + \sqrt{5}(3) + \sqrt{5}(-\sqrt{2}) + \sqrt{5}(-\sqrt{5})\)[/tex]
Group like terms together:
[tex]\[ = 3(3) + 3(-\sqrt{2}) + 3(-\sqrt{5}) + 3\sqrt{2} + \sqrt{2}(-\sqrt{2}) + \sqrt{2}(-\sqrt{5}) + 3\sqrt{5} + \sqrt{5}(-\sqrt{2}) + \sqrt{5}(-\sqrt{5}) \][/tex]
Simplify terms involving square roots:
[tex]\[ = 9 - 3\sqrt{2} - 3\sqrt{5} + 3\sqrt{2} - 2 -\sqrt{10} + 3\sqrt{5} - \sqrt{10} - 5 \][/tex]
Combine and cancel out terms:
[tex]\[ = 9 - 2 - 5 - 2\sqrt{10} = 2 - 2\sqrt{10} \][/tex]
5. Combine the simplified numerator and denominator:
[tex]\[ \frac{6 - 2\sqrt{2} - 2\sqrt{5}}{2 - 2\sqrt{10}} = \frac{2(3 - \sqrt{2} - \sqrt{5})}{2(1 - \sqrt{10})} = \frac{3 - \sqrt{2} - \sqrt{5}}{1 - \sqrt{10}} \][/tex]
So, the final result is:
[tex]\(\frac{3 - \sqrt{2} - \sqrt{5}}{1 - \sqrt{10}}\)[/tex]
To achieve this, we will multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of a sum of square roots like [tex]\(3 + \sqrt{2} + \sqrt{5}\)[/tex] is usually taken by changing the signs of the square roots, so in this case, it will be [tex]\(3 - \sqrt{2} - \sqrt{5}\)[/tex].
Here are the steps in detail:
1. Identify the original expression:
[tex]\[ \frac{2}{3+\sqrt{2}+\sqrt{5}} \][/tex]
2. Multiply both the numerator and the denominator by the conjugate of the denominator:
[tex]\[ \frac{2}{3+\sqrt{2}+\sqrt{5}} \times \frac{3-\sqrt{2}-\sqrt{5}}{3-\sqrt{2}-\sqrt{5}} = \frac{2(3-\sqrt{2}-\sqrt{5})}{(3+\sqrt{2}+\sqrt{5})(3-\sqrt{2}-\sqrt{5})} \][/tex]
3. Expand the numerator:
[tex]\[ 2(3 - \sqrt{2} - \sqrt{5}) = 6 - 2\sqrt{2} - 2\sqrt{5} \][/tex]
4. Expand the denominator using the distributive property (a-b)(a+b) to handle the square roots:
[tex]\[ (3+\sqrt{2}+\sqrt{5})(3-\sqrt{2}-\sqrt{5}) \][/tex]
Let’s break it down using the distributive property:
[tex]\[ (3+\sqrt{2}+\sqrt{5})(3-\sqrt{2}-\sqrt{5}) \][/tex]
= [tex]\(3(3) + 3(-\sqrt{2}) + 3(-\sqrt{5}) + \sqrt{2}(3) + \sqrt{2}(-\sqrt{2}) + \sqrt{2}(-\sqrt{5}) + \sqrt{5}(3) + \sqrt{5}(-\sqrt{2}) + \sqrt{5}(-\sqrt{5})\)[/tex]
Group like terms together:
[tex]\[ = 3(3) + 3(-\sqrt{2}) + 3(-\sqrt{5}) + 3\sqrt{2} + \sqrt{2}(-\sqrt{2}) + \sqrt{2}(-\sqrt{5}) + 3\sqrt{5} + \sqrt{5}(-\sqrt{2}) + \sqrt{5}(-\sqrt{5}) \][/tex]
Simplify terms involving square roots:
[tex]\[ = 9 - 3\sqrt{2} - 3\sqrt{5} + 3\sqrt{2} - 2 -\sqrt{10} + 3\sqrt{5} - \sqrt{10} - 5 \][/tex]
Combine and cancel out terms:
[tex]\[ = 9 - 2 - 5 - 2\sqrt{10} = 2 - 2\sqrt{10} \][/tex]
5. Combine the simplified numerator and denominator:
[tex]\[ \frac{6 - 2\sqrt{2} - 2\sqrt{5}}{2 - 2\sqrt{10}} = \frac{2(3 - \sqrt{2} - \sqrt{5})}{2(1 - \sqrt{10})} = \frac{3 - \sqrt{2} - \sqrt{5}}{1 - \sqrt{10}} \][/tex]
So, the final result is:
[tex]\(\frac{3 - \sqrt{2} - \sqrt{5}}{1 - \sqrt{10}}\)[/tex]
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.
