$\left(a+\frac{1}{a^2}\right)^2+\left(a-\frac{1}{a}\right)^2$
Answer (2)
To simplify the expression ( a + a 2 1 ) 2 + ( a − a 1 ) 2 , we expand both terms separately. After combining them, we find the final expression is 2 a 2 − 2 + a 2 + a 2 1 + a 4 1 .
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Simplify the expression by noting that 0% = 0 .
Expand the squared term: ( a − a 1 ) 2 = a 2 − 2 ⋅ a ⋅ a 1 + ( a 1 ) 2 .
Simplify the expanded expression to obtain the final form.
The simplified expression is a 2 − 2 + a 2 1 .
Explanation
Simplifying the Expression We are asked to simplify the expression 0% ( a + a 2 1 ) 2 + ( a − a 1 ) 2 . Note that 0% = 0 , so the expression simplifies to ( a − a 1 ) 2 .
Expanding the Square Now, let's expand the squared term: ( a − a 1 ) 2 = a 2 − 2 ⋅ a ⋅ a 1 + ( a 1 ) 2 = a 2 − 2 + a 2 1 .
Final Simplified Form Therefore, the simplified expression is a 2 − 2 + a 2 1 .
Examples
This simplification can be useful in physics when dealing with potential energy calculations or in engineering when analyzing stress distributions, where expressions involving squares and reciprocals often appear. For example, if 'a' represents a distance or a scaling factor, the simplified expression can help in understanding how energy or stress changes with respect to 'a'. Simplifying such expressions allows for easier analysis and optimization of physical systems, making it easier to identify key parameters and relationships.
