IDNLearn.com provides a seamless experience for finding accurate answers. Join our knowledgeable community to find the answers you need for any topic or issue.
Sagot :
To simplify the expression [tex]\(\frac{2}{\sqrt{7}+\sqrt{2}} + \frac{1}{\sqrt{7}-\sqrt{2}}\)[/tex] and write it as a single fraction, we will follow these steps:
1. Rationalize the denominators:
Rationalizing the denominator means multiplying the numerator and denominator of each fraction by the conjugate of the denominator.
2. Simplify each fraction:
After rationalizing, simplify each fraction separately.
3. Combine the fractions:
Express both fractions with a common denominator and combine them.
Let's begin by rationalizing the denominators:
### Step 1: Rationalize the denominators
#### For [tex]\(\frac{2}{\sqrt{7} + \sqrt{2}}\)[/tex]:
Multiply the numerator and the denominator by the conjugate of the denominator, which is [tex]\(\sqrt{7} - \sqrt{2}\)[/tex]:
[tex]\[ \frac{2}{\sqrt{7}+\sqrt{2}} \cdot \frac{\sqrt{7}-\sqrt{2}}{\sqrt{7}-\sqrt{2}} = \frac{2(\sqrt{7}-\sqrt{2})}{(\sqrt{7}+\sqrt{2})(\sqrt{7}-\sqrt{2})} \][/tex]
The denominator becomes:
[tex]\[ (\sqrt{7} + \sqrt{2})(\sqrt{7} - \sqrt{2}) = 7 - 2 = 5 \][/tex]
So, the fraction simplifies to:
[tex]\[ \frac{2(\sqrt{7} - \sqrt{2})}{5} = \frac{2\sqrt{7} - 2\sqrt{2}}{5} \][/tex]
#### For [tex]\(\frac{1}{\sqrt{7} - \sqrt{2}}\)[/tex]:
Multiply the numerator and the denominator by the conjugate of the denominator, which is [tex]\(\sqrt{7} + \sqrt{2}\)[/tex]:
[tex]\[ \frac{1}{\sqrt{7} - \sqrt{2}} \cdot \frac{\sqrt{7}+\sqrt{2}}{\sqrt{7}+\sqrt{2}} = \frac{\sqrt{7}+\sqrt{2}}{(\sqrt{7}-\sqrt{2})(\sqrt{7}+\sqrt{2})} \][/tex]
The denominator becomes:
[tex]\[ (\sqrt{7} - \sqrt{2})(\sqrt{7} + \sqrt{2}) = 7 - 2 = 5 \][/tex]
So, the fraction simplifies to:
[tex]\[ \frac{\sqrt{7} + \sqrt{2}}{5} \][/tex]
### Step 2: Combine the fractions
Now we add the two simplified fractions:
[tex]\[ \frac{2\sqrt{7} - 2\sqrt{2}}{5} + \frac{\sqrt{7} + \sqrt{2}}{5} \][/tex]
Combine them over a common denominator:
[tex]\[ \frac{(2\sqrt{7} - 2\sqrt{2}) + (\sqrt{7} + \sqrt{2})}{5} \][/tex]
Combine the like terms in the numerator:
[tex]\[ \frac{2\sqrt{7} + \sqrt{7} - 2\sqrt{2} + \sqrt{2}}{5} = \frac{3\sqrt{7} - \sqrt{2}}{5} \][/tex]
### Step 3: Conclusion
The expression [tex]\(\frac{2}{\sqrt{7}+\sqrt{2}} + \frac{1}{\sqrt{7}-\sqrt{2}}\)[/tex] as a single fraction is:
[tex]\[ \boxed{\frac{3\sqrt{7} - \sqrt{2}}{5}} \][/tex]
1. Rationalize the denominators:
Rationalizing the denominator means multiplying the numerator and denominator of each fraction by the conjugate of the denominator.
2. Simplify each fraction:
After rationalizing, simplify each fraction separately.
3. Combine the fractions:
Express both fractions with a common denominator and combine them.
Let's begin by rationalizing the denominators:
### Step 1: Rationalize the denominators
#### For [tex]\(\frac{2}{\sqrt{7} + \sqrt{2}}\)[/tex]:
Multiply the numerator and the denominator by the conjugate of the denominator, which is [tex]\(\sqrt{7} - \sqrt{2}\)[/tex]:
[tex]\[ \frac{2}{\sqrt{7}+\sqrt{2}} \cdot \frac{\sqrt{7}-\sqrt{2}}{\sqrt{7}-\sqrt{2}} = \frac{2(\sqrt{7}-\sqrt{2})}{(\sqrt{7}+\sqrt{2})(\sqrt{7}-\sqrt{2})} \][/tex]
The denominator becomes:
[tex]\[ (\sqrt{7} + \sqrt{2})(\sqrt{7} - \sqrt{2}) = 7 - 2 = 5 \][/tex]
So, the fraction simplifies to:
[tex]\[ \frac{2(\sqrt{7} - \sqrt{2})}{5} = \frac{2\sqrt{7} - 2\sqrt{2}}{5} \][/tex]
#### For [tex]\(\frac{1}{\sqrt{7} - \sqrt{2}}\)[/tex]:
Multiply the numerator and the denominator by the conjugate of the denominator, which is [tex]\(\sqrt{7} + \sqrt{2}\)[/tex]:
[tex]\[ \frac{1}{\sqrt{7} - \sqrt{2}} \cdot \frac{\sqrt{7}+\sqrt{2}}{\sqrt{7}+\sqrt{2}} = \frac{\sqrt{7}+\sqrt{2}}{(\sqrt{7}-\sqrt{2})(\sqrt{7}+\sqrt{2})} \][/tex]
The denominator becomes:
[tex]\[ (\sqrt{7} - \sqrt{2})(\sqrt{7} + \sqrt{2}) = 7 - 2 = 5 \][/tex]
So, the fraction simplifies to:
[tex]\[ \frac{\sqrt{7} + \sqrt{2}}{5} \][/tex]
### Step 2: Combine the fractions
Now we add the two simplified fractions:
[tex]\[ \frac{2\sqrt{7} - 2\sqrt{2}}{5} + \frac{\sqrt{7} + \sqrt{2}}{5} \][/tex]
Combine them over a common denominator:
[tex]\[ \frac{(2\sqrt{7} - 2\sqrt{2}) + (\sqrt{7} + \sqrt{2})}{5} \][/tex]
Combine the like terms in the numerator:
[tex]\[ \frac{2\sqrt{7} + \sqrt{7} - 2\sqrt{2} + \sqrt{2}}{5} = \frac{3\sqrt{7} - \sqrt{2}}{5} \][/tex]
### Step 3: Conclusion
The expression [tex]\(\frac{2}{\sqrt{7}+\sqrt{2}} + \frac{1}{\sqrt{7}-\sqrt{2}}\)[/tex] as a single fraction is:
[tex]\[ \boxed{\frac{3\sqrt{7} - \sqrt{2}}{5}} \][/tex]
We appreciate your contributions to this forum. Don't forget to check back for the latest answers. Keep asking, answering, and sharing useful information. Your search for solutions ends at IDNLearn.com. Thank you for visiting, and we look forward to helping you again.
