Question
Rationalize the denominator. \( \frac{7}{1+\sqrt{7}} \) (Simplify your answer. Type an exact answer, using radicals as needed.) (iver
Ask by Huang Lyons. in the United States
Mar 14,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The rationalized form is \( \frac{7\sqrt{7}-7}{6} \).
Solution
Calculate or simplify the expression \( \frac{7}{1+\sqrt{7}} \).
Calculate the value by following steps:
- step0: Calculate:
\(\frac{7}{1+\sqrt{7}}\)
- step1: Multiply by the Conjugate:
\(\frac{7\left(1-\sqrt{7}\right)}{\left(1+\sqrt{7}\right)\left(1-\sqrt{7}\right)}\)
- step2: Multiply the numbers:
\(\frac{7\left(1-\sqrt{7}\right)}{-6}\)
- step3: Multiply the numbers:
\(\frac{7-7\sqrt{7}}{-6}\)
- step4: Calculate:
\(\frac{7\sqrt{7}-7}{6}\)
The rationalized form of \( \frac{7}{1+\sqrt{7}} \) is \( \frac{7\sqrt{7}-7}{6} \).
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Simplify this solution The Deep Dive
To rationalize the denominator of \( \frac{7}{1+\sqrt{7}} \), multiply the numerator and the denominator by the conjugate of the denominator, which is \( 1-\sqrt{7} \): \[ \frac{7(1-\sqrt{7})}{(1+\sqrt{7})(1-\sqrt{7})} \] Now, the denominator simplifies using the difference of squares: \[ (1+\sqrt{7})(1-\sqrt{7}) = 1^2 - (\sqrt{7})^2 = 1 - 7 = -6 \] Thus, the expression becomes: \[ \frac{7(1-\sqrt{7})}{-6} \] Now simplify the numerator: \[ = \frac{7 - 7\sqrt{7}}{-6} = \frac{-7 + 7\sqrt{7}}{6} \] So, the rationalized form of the expression is: \[ \frac{7\sqrt{7} - 7}{6} \]
