Multiply. Simplify if possible.
[tex]$(3+\sqrt{7})(3-\sqrt{7}) = \square$[/tex]
Answer (2)
Recognize the expression as a difference of squares: ( a + b ) ( a − b ) .
Apply the difference of squares formula: ( a + b ) ( a − b ) = a 2 − b 2 .
Calculate the squares: 3 2 = 9 and ( 7 ) 2 = 7 .
Subtract to find the simplified expression: 9 − 7 = 2 .
Explanation
Recognizing the Pattern We are asked to multiply and simplify the expression ( 3 + 7 ) ( 3 − 7 ) . This expression is in the form of ( a + b ) ( a − b ) , which is a difference of squares.
Applying the Difference of Squares The difference of squares formula is ( a + b ) ( a − b ) = a 2 − b 2 . In our case, a = 3 and b = 7 .
Calculating the Squares Now, we calculate a 2 and b 2 . We have a 2 = 3 2 = 9 and b 2 = ( 7 ) 2 = 7 .
Simplifying the Expression Finally, we subtract b 2 from a 2 : 9 − 7 = 2 . Therefore, ( 3 + 7 ) ( 3 − 7 ) = 2 .
Final Answer The simplified expression is 2.
Examples
The difference of squares pattern is useful in various fields, such as engineering and physics, where simplifying expressions can make calculations easier. For example, when calculating the area of a region or solving equations involving square roots, recognizing and applying the difference of squares can significantly reduce the complexity of the problem. This pattern also forms the basis for more advanced algebraic techniques.
The expression ( 3 + 7 ) ( 3 − 7 ) simplifies to 2 by applying the difference of squares formula. We find a 2 = 9 and b 2 = 7 , leading to the final result of 9 − 7 = 2 .
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