Question
omplete the following Simplify \( \left(x+\frac{1}{x}\right)^{2} \)
Ask by Colon Turner. in South Africa
Jan 26,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\[
\left(x + \frac{1}{x}\right)^{2} = x^2 + 2 + \frac{1}{x^2}
\]
Solution
Certainly! Let's simplify the expression:
\[
\left(x + \frac{1}{x}\right)^{2}
\]
**Step 1: Expand the Square**
Use the formula \((a + b)^2 = a^2 + 2ab + b^2\):
\[
\left(x + \frac{1}{x}\right)^{2} = x^2 + 2 \cdot x \cdot \frac{1}{x} + \left(\frac{1}{x}\right)^2
\]
**Step 2: Simplify Each Term**
- \(x^2\) remains as is.
- \(2 \cdot x \cdot \frac{1}{x} = 2\) (since \(x \cdot \frac{1}{x} = 1\))
- \(\left(\frac{1}{x}\right)^2 = \frac{1}{x^2}\)
**Step 3: Combine the Terms**
\[
x^2 + 2 + \frac{1}{x^2}
\]
**Final Simplified Form:**
\[
\left(x + \frac{1}{x}\right)^{2} = x^2 + 2 + \frac{1}{x^2}
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Mind Expander
To simplify \( \left(x+\frac{1}{x}\right)^{2} \), we can use the identity for the square of a binomial. This gives us: \[ \left(x+\frac{1}{x}\right)^{2} = x^{2} + 2\left(x \cdot \frac{1}{x}\right) + \left(\frac{1}{x}\right)^{2} \] This simplifies further to: \[ = x^{2} + 2 + \frac{1}{x^{2}} \] So, the final simplified form is: \[ x^{2} + 2 + \frac{1}{x^{2}} \]
