Simplify $\frac{54 k^2-6}{3 k+1}$
Answer (2)
Factor out the greatest common divisor from the numerator: 54 k 2 − 6 = 6 ( 9 k 2 − 1 ) .
Factor the numerator as a difference of squares: 6 ( 9 k 2 − 1 ) = 6 ( 3 k − 1 ) ( 3 k + 1 ) .
Simplify the expression by cancelling common factors: 3 k + 1 6 ( 3 k − 1 ) ( 3 k + 1 ) = 6 ( 3 k − 1 ) .
Distribute to obtain the final simplified expression: 6 ( 3 k − 1 ) = 18 k − 6 . The simplified expression is 18 k − 6 .
Explanation
Understanding the Problem We are asked to simplify the expression 3 k + 1 54 k 2 − 6 .
Factoring out the GCD First, we factor out the greatest common divisor (GCD) from the numerator, which is 6: 54 k 2 − 6 = 6 ( 9 k 2 − 1 ) .
Factoring the Difference of Squares Next, we recognize that the expression inside the parenthesis is a difference of squares: 9 k 2 − 1 = ( 3 k ) 2 − 1 2 . We can factor this as ( 3 k − 1 ) ( 3 k + 1 ) . Therefore, the numerator becomes 6 ( 3 k − 1 ) ( 3 k + 1 ) .
Rewriting the Expression Now, we can rewrite the original expression as: 3 k + 1 6 ( 3 k − 1 ) ( 3 k + 1 ) .
Cancelling Common Factors We can cancel the common factor of ( 3 k + 1 ) from the numerator and the denominator, provided that 3 k + 1 = 0 : 3 k + 1 6 ( 3 k − 1 ) ( 3 k + 1 ) = 6 ( 3 k − 1 ) .
Distributing the Constant Finally, we distribute the 6 to get the simplified expression: 6 ( 3 k − 1 ) = 18 k − 6 .
Examples
Simplifying rational expressions is a fundamental skill in algebra and is used in various real-world applications. For instance, in physics, you might encounter rational expressions when dealing with electrical circuits or fluid dynamics. Simplifying these expressions can make calculations easier and provide a clearer understanding of the relationships between different variables. In engineering, simplifying such expressions can help in optimizing designs and reducing computational complexity.
To simplify the expression 3 k + 1 54 k 2 − 6 , we factor the numerator to get 6 ( 3 k − 1 ) ( 3 k + 1 ) and cancel out the common factor 3 k + 1 . This leads to the final simplified result of 18 k − 6 .
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