At Westonci.ca, we make it easy for you to get the answers you need from a community of knowledgeable individuals. Join our platform to connect with experts ready to provide precise answers to your questions in different areas. 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]
Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.
