$\left(a+\frac{1}{a^7}\right)^{2^2}+\left(a-\frac{1}{a}\right)^2$
Answer (2)
The expression ( a + a 7 1 ) 4 + ( a − a 1 ) 2 simplifies to a 4 + a 2 − 2 + a 2 1 + a 4 4 + a 12 6 + a 20 4 + a 28 1 .
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Simplify the exponent: 2 2 = 4 .
Expand the first term: ( a + a 7 1 ) 4 = a 4 + a 4 4 + a 12 6 + a 20 4 + a 28 1 .
Expand the second term: ( a − a 1 ) 2 = a 2 − 2 + a 2 1 .
Combine the terms to get the simplified expression: a 4 + a 2 − 2 + a 2 1 + a 4 4 + a 12 6 + a 20 4 + a 28 1 = a 28 a 26 ( a 2 − 1 ) 2 + ( a 8 + 1 ) 4 .
Explanation
Understanding the Problem We are given the expression ( a + a 7 1 ) 2 2 + ( a − a 1 ) 2 and we want to simplify it.
Simplifying the Exponent First, we simplify the exponent 2 2 = 4 . So the expression becomes ( a + a 7 1 ) 4 + ( a − a 1 ) 2 .
Expanding the First Term Next, we expand the first term ( a + a 7 1 ) 4 using the binomial theorem:
( a + a 7 1 ) 4 = a 4 + 4 a 3 ( a 7 1 ) + 6 a 2 ( a 7 1 ) 2 + 4 a ( a 7 1 ) 3 + ( a 7 1 ) 4
= a 4 + a 7 4 a 3 + a 14 6 a 2 + a 21 4 a + a 28 1
= a 4 + a 4 4 + a 12 6 + a 20 4 + a 28 1
Expanding the Second Term Now, we expand the second term ( a − a 1 ) 2 :
( a − a 1 ) 2 = a 2 − 2 a ( a 1 ) + ( a 1 ) 2
= a 2 − 2 + a 2 1
Adding the Expanded Terms We add the two expanded terms:
a 4 + a 4 4 + a 12 6 + a 20 4 + a 28 1 + a 2 − 2 + a 2 1
= a 4 + a 2 − 2 + a 2 1 + a 4 4 + a 12 6 + a 20 4 + a 28 1
Final Simplified Expression Therefore, the simplified expression is:
a 4 + a 2 − 2 + a 2 1 + a 4 4 + a 12 6 + a 20 4 + a 28 1
Alternatively, we can write it as a single fraction:
a 28 a 26 ( a 2 − 1 ) 2 + ( a 8 + 1 ) 4
Examples
Simplifying algebraic expressions is a fundamental skill in mathematics with applications in various fields. For instance, in physics, simplifying expressions can help in analyzing complex systems, such as electrical circuits or mechanical systems. In computer graphics, simplifying expressions can optimize rendering algorithms, leading to faster and more efficient graphics processing. Moreover, in economics, simplifying expressions can aid in modeling and analyzing market trends, making it easier to understand and predict economic behavior.
