Asked by woord
Simplify. Rationalize the denominator.
4
–
7+
square root 2
4
–
7+
square root 2
Answers
Answered by
GPT-4o mini
To simplify the expression \( \frac{4}{7 + \sqrt{2}} \) and rationalize the denominator, we can multiply the numerator and the denominator by the conjugate of the denominator.
The conjugate of \( 7 + \sqrt{2} \) is \( 7 - \sqrt{2} \).
So we multiply the fraction by \( \frac{7 - \sqrt{2}}{7 - \sqrt{2}} \):
\[
\frac{4}{7 + \sqrt{2}} \cdot \frac{7 - \sqrt{2}}{7 - \sqrt{2}} = \frac{4(7 - \sqrt{2})}{(7 + \sqrt{2})(7 - \sqrt{2})}
\]
Now, calculate the denominator using the difference of squares:
\[
(7 + \sqrt{2})(7 - \sqrt{2}) = 7^2 - (\sqrt{2})^2 = 49 - 2 = 47
\]
Next, calculate the numerator:
\[
4(7 - \sqrt{2}) = 28 - 4\sqrt{2}
\]
Putting it all together, we have:
\[
\frac{28 - 4\sqrt{2}}{47}
\]
Therefore, the simplified expression with the denominator rationalized is:
\[
\frac{28 - 4\sqrt{2}}{47}
\]
The conjugate of \( 7 + \sqrt{2} \) is \( 7 - \sqrt{2} \).
So we multiply the fraction by \( \frac{7 - \sqrt{2}}{7 - \sqrt{2}} \):
\[
\frac{4}{7 + \sqrt{2}} \cdot \frac{7 - \sqrt{2}}{7 - \sqrt{2}} = \frac{4(7 - \sqrt{2})}{(7 + \sqrt{2})(7 - \sqrt{2})}
\]
Now, calculate the denominator using the difference of squares:
\[
(7 + \sqrt{2})(7 - \sqrt{2}) = 7^2 - (\sqrt{2})^2 = 49 - 2 = 47
\]
Next, calculate the numerator:
\[
4(7 - \sqrt{2}) = 28 - 4\sqrt{2}
\]
Putting it all together, we have:
\[
\frac{28 - 4\sqrt{2}}{47}
\]
Therefore, the simplified expression with the denominator rationalized is:
\[
\frac{28 - 4\sqrt{2}}{47}
\]