Certainly! Let's rationalize the denominator of the fraction [tex]\(\frac{1}{2+\sqrt{7}}\)[/tex] step by step.
### Step 1: Identify the Conjugate
The first step is to identify the conjugate of the denominator [tex]\(2 + \sqrt{7}\)[/tex]. The conjugate of [tex]\(2 + \sqrt{7}\)[/tex] is [tex]\(2 - \sqrt{7}\)[/tex]. Multiplying by the conjugate can help us eliminate the square root in the denominator.
### Step 2: Multiply the Fraction
We multiply both the numerator and the denominator by the conjugate of the denominator:
[tex]\[
\frac{1}{2+\sqrt{7}} \times \frac{2-\sqrt{7}}{2-\sqrt{7}} = \frac{1 \cdot (2 - \sqrt{7})}{(2 + \sqrt{7})(2 - \sqrt{7})}
\][/tex]
### Step 3: Simplify the Numerator
Simplify the numerator:
[tex]\[
1 \cdot (2 - \sqrt{7}) = 2 - \sqrt{7}
\][/tex]
### Step 4: Simplify the Denominator
Now, simplify the denominator by using the difference of squares formula:
[tex]\[
(2 + \sqrt{7})(2 - \sqrt{7}) = 2^2 - (\sqrt{7})^2 = 4 - 7 = -3
\][/tex]
### Step 5: Write the Simplified Fraction
Putting the simplified numerator and denominator together, we have:
[tex]\[
\frac{2 - \sqrt{7}}{-3}
\][/tex]
### Step 6: Separate the Fraction
You can also separate the fraction into two parts for clarity:
[tex]\[
\frac{2 - \sqrt{7}}{-3} = \frac{2}{-3} - \frac{\sqrt{7}}{-3} = -\frac{2}{3} + \frac{\sqrt{7}}{3}
\][/tex]
### Final Answer
Thus, the rationalized form of [tex]\(\frac{1}{2+\sqrt{7}}\)[/tex] is:
[tex]\[
-\frac{2}{3} + \frac{\sqrt{7}}{3}
\][/tex]
Or equivalently, in its combined form:
[tex]\[
\frac{2 - \sqrt{7}}{-3}
\][/tex]
Additionally, if you would like to know the approximate decimal values:
[tex]\[
2 - \sqrt{7} \approx -0.6457513110645907
\][/tex]
[tex]\[
\frac{2 - \sqrt{7}}{-3} \approx 0.21525043702153024
\][/tex]
These steps provide a detailed process for rationalizing the denominator of [tex]\(\frac{1}{2+\sqrt{7}}\)[/tex].
