IDNLearn.com: Where curiosity meets clarity and questions find their answers. Ask your questions and get detailed, reliable answers from our community of knowledgeable experts.
Sagot :
To simplify the expression [tex]\(\frac{2}{\sqrt{5}+\sqrt{3}+2}\)[/tex], we aim to rationalize the denominator. This involves eliminating the square roots in the denominator. Here's a detailed step-by-step solution:
1. Express the Problem:
[tex]\[ \frac{2}{\sqrt{5} + \sqrt{3} + 2} \][/tex]
2. Multiply by the Conjugate:
The goal in rationalizing the denominator is to clear out the square roots. However, the denominator has two irrational terms, [tex]\(\sqrt{5}\)[/tex] and [tex]\(\sqrt{3}\)[/tex], plus a rational term, [tex]\(2\)[/tex]. To rationalize such a complex denominator, we can use a method involving the sum and differences of squares.
Consider the conjugate of the whole denominator by creating a group with similar terms, say like [tex]\( (\sqrt{5} + \sqrt{3}) + 2\)[/tex]. We should multiply the numerator and the denominator by the expression containing both irrational parts as negative, which would be:
[tex]\[ \sqrt{5} + \sqrt{3} - 2 \][/tex]
3. Construct the Rationalizing Term and Multiply:
So, we multiply the numerator and the denominator by [tex]\((\sqrt{5} + \sqrt{3} - 2)\)[/tex]:
[tex]\[ \frac{2}{\sqrt{5} + \sqrt{3} + 2} \times \frac{\sqrt{5} + \sqrt{3} - 2}{\sqrt{5} + \sqrt{3} - 2} \][/tex]
This gives us:
[tex]\[ \frac{2 (\sqrt{5} + \sqrt{3} - 2)}{(\sqrt{5} + \sqrt{3} + 2)(\sqrt{5} + \sqrt{3} - 2)} \][/tex]
4. Simplify the Denominator (Use Difference of Squares):
To simplify the denominator, recall the formula [tex]\((a + b)(a - b) = a^2 - b^2\)[/tex]:
Let [tex]\(a = \sqrt{5} + \sqrt{3}\)[/tex] and [tex]\(b = 2\)[/tex], then:
[tex]\[ (\sqrt{5} + \sqrt{3})^2 - (2)^2 \][/tex]
Calculate the values:
[tex]\[ (\sqrt{5} + \sqrt{3})^2 = (\sqrt{5})^2 + 2(\sqrt{5})(\sqrt{3}) + (\sqrt{3})^2 = 5 + 2\sqrt{15} + 3 = 8 + 2\sqrt{15} \][/tex]
[tex]\[ (2)^2 = 4 \][/tex]
Substitute these into the expression:
[tex]\[ (\sqrt{5} + \sqrt{3} + 2)(\sqrt{5} + \sqrt{3} - 2) = (8 + 2\sqrt{15}) - 4 = 4 + 2\sqrt{15} \][/tex]
5. Put the Rationalized Expression Together:
Having rationalised the denominator:
[tex]\[ \frac{2 (\sqrt{5} + \sqrt{3} - 2)}{4 + 2\sqrt{15}} \][/tex]
6. Further Simplify the Expression:
To make simplification easier, factor the denominator:
[tex]\[ 4 + 2\sqrt{15} = 2(2 + \sqrt{15}) \][/tex]
Therefore, the expression becomes:
[tex]\[ \frac{2 (\sqrt{5} + \sqrt{3} - 2)}{2(2 + \sqrt{15})} = \frac{\sqrt{5} + \sqrt{3} - 2}{2 + \sqrt{15}} \][/tex]
Thus, the rationalized simplified form of the given expression is:
[tex]\[ \boxed{\frac{\sqrt{5} + \sqrt{3} - 2}{2 + \sqrt{15}}} \][/tex]
1. Express the Problem:
[tex]\[ \frac{2}{\sqrt{5} + \sqrt{3} + 2} \][/tex]
2. Multiply by the Conjugate:
The goal in rationalizing the denominator is to clear out the square roots. However, the denominator has two irrational terms, [tex]\(\sqrt{5}\)[/tex] and [tex]\(\sqrt{3}\)[/tex], plus a rational term, [tex]\(2\)[/tex]. To rationalize such a complex denominator, we can use a method involving the sum and differences of squares.
Consider the conjugate of the whole denominator by creating a group with similar terms, say like [tex]\( (\sqrt{5} + \sqrt{3}) + 2\)[/tex]. We should multiply the numerator and the denominator by the expression containing both irrational parts as negative, which would be:
[tex]\[ \sqrt{5} + \sqrt{3} - 2 \][/tex]
3. Construct the Rationalizing Term and Multiply:
So, we multiply the numerator and the denominator by [tex]\((\sqrt{5} + \sqrt{3} - 2)\)[/tex]:
[tex]\[ \frac{2}{\sqrt{5} + \sqrt{3} + 2} \times \frac{\sqrt{5} + \sqrt{3} - 2}{\sqrt{5} + \sqrt{3} - 2} \][/tex]
This gives us:
[tex]\[ \frac{2 (\sqrt{5} + \sqrt{3} - 2)}{(\sqrt{5} + \sqrt{3} + 2)(\sqrt{5} + \sqrt{3} - 2)} \][/tex]
4. Simplify the Denominator (Use Difference of Squares):
To simplify the denominator, recall the formula [tex]\((a + b)(a - b) = a^2 - b^2\)[/tex]:
Let [tex]\(a = \sqrt{5} + \sqrt{3}\)[/tex] and [tex]\(b = 2\)[/tex], then:
[tex]\[ (\sqrt{5} + \sqrt{3})^2 - (2)^2 \][/tex]
Calculate the values:
[tex]\[ (\sqrt{5} + \sqrt{3})^2 = (\sqrt{5})^2 + 2(\sqrt{5})(\sqrt{3}) + (\sqrt{3})^2 = 5 + 2\sqrt{15} + 3 = 8 + 2\sqrt{15} \][/tex]
[tex]\[ (2)^2 = 4 \][/tex]
Substitute these into the expression:
[tex]\[ (\sqrt{5} + \sqrt{3} + 2)(\sqrt{5} + \sqrt{3} - 2) = (8 + 2\sqrt{15}) - 4 = 4 + 2\sqrt{15} \][/tex]
5. Put the Rationalized Expression Together:
Having rationalised the denominator:
[tex]\[ \frac{2 (\sqrt{5} + \sqrt{3} - 2)}{4 + 2\sqrt{15}} \][/tex]
6. Further Simplify the Expression:
To make simplification easier, factor the denominator:
[tex]\[ 4 + 2\sqrt{15} = 2(2 + \sqrt{15}) \][/tex]
Therefore, the expression becomes:
[tex]\[ \frac{2 (\sqrt{5} + \sqrt{3} - 2)}{2(2 + \sqrt{15})} = \frac{\sqrt{5} + \sqrt{3} - 2}{2 + \sqrt{15}} \][/tex]
Thus, the rationalized simplified form of the given expression is:
[tex]\[ \boxed{\frac{\sqrt{5} + \sqrt{3} - 2}{2 + \sqrt{15}}} \][/tex]
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. Thank you for choosing IDNLearn.com for your queries. We’re here to provide accurate answers, so visit us again soon.
