\(a+\frac{1}{a^2}\)^2+\left(a-\frac{1}{a}\right)^2
Answer (2)
To simplify a + a 2 1 ^2 + ( a − a 1 ) ^2, we expand each term using the square of a binomial formula. The result becomes 2 a 2 − 2 + a 4 1 + a 2 1 + a 2 . Hence, the final simplified expression encapsulates components from both terms combined.
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Recognize that 0% = 0 , so the first term becomes 0.
Expand the second term: ( a − a 1 ) 2 = a 2 − 2 + a 2 1 .
Combine the terms to get the simplified expression.
The simplified expression is a 2 − 2 + a 2 1 .
Explanation
Understanding the Expression We are asked to simplify the expression 0% ( a + a 2 1 ) 2 + ( a − a 1 ) 2 . Note that 0% is equivalent to 0 . Therefore, the expression simplifies to 0 × ( a + a 2 1 ) 2 + ( a − a 1 ) 2 .
Simplifying the First Term Since anything multiplied by 0 is 0, the first term becomes 0. Thus, we have 0 + ( a − a 1 ) 2 .
Expanding the Second Term Now we need to expand the second term: ( a − a 1 ) 2 = a 2 − 2 ⋅ a ⋅ a 1 + ( a 1 ) 2 = a 2 − 2 + a 2 1 .
Final Simplified Expression Therefore, the simplified expression is a 2 − 2 + a 2 1 .
Examples
Understanding how to simplify algebraic expressions is crucial in many fields, such as physics and engineering. For example, when analyzing the motion of a projectile, you might encounter complex equations involving variables like initial velocity and launch angle. Simplifying these equations allows you to more easily predict the projectile's trajectory and range. Similarly, in electrical engineering, simplifying circuit equations can help you determine the current and voltage at different points in the circuit, making it easier to design and troubleshoot electronic devices.
