$r^2+\frac{9}{r^2}-6$
Answer (2)
The problem involves simplifying the expression r 2 + r 2 9 − 6 .
Rewrite the expression as r 2 − 6 + r 2 9 .
Recognize the expression as a perfect square.
Factor the expression as ( r − r 3 ) 2 .
The simplified expression is ( r − r 3 ) 2 .
Explanation
Understanding the Expression We are given the expression r 2 + r 2 9 − 6 and we want to simplify it.
Rewriting the Expression We can rewrite the expression as r 2 − 6 + r 2 9 . Notice that this looks similar to the expansion of a squared term.
Recognizing the Square Recall that ( a − b ) 2 = a 2 − 2 ab + b 2 . Let's try to express our given expression in this form. We can rewrite r 2 − 6 + r 2 9 as ( r ) 2 − 2 ( r ) ( r 3 ) + ( r 3 ) 2 .
Factoring the Expression Now we can see that this is indeed a perfect square: ( r − r 3 ) 2 = r 2 − 2 ( r ) ( r 3 ) + ( r 3 ) 2 = r 2 − 6 + r 2 9 .
Final Answer Therefore, the simplified expression is ( r − r 3 ) 2 .
Examples
This type of algebraic simplification is useful in various fields, such as physics and engineering, where complex equations can be simplified to make them easier to analyze and solve. For example, in mechanics, simplifying expressions involving kinetic energy or potential energy can help in understanding the behavior of systems. Similarly, in electrical engineering, simplifying expressions involving impedance or admittance can aid in circuit analysis. By recognizing patterns and applying algebraic identities, we can transform complex expressions into simpler forms that are easier to work with.
The expression r 2 + r 2 9 − 6 can be simplified by recognizing it as a perfect square. The simplified form is ( r − r 3 ) 2 . Therefore, the final simplified expression is ( r − r 3 ) 2 .
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