Multiply: $(\sqrt{10}+2 \sqrt{8})(\sqrt{10}-2 \sqrt{8})$
Answer (2)
The expression ( 10 + 2 8 ) ( 10 − 2 8 ) simplifies to − 22 using the difference of squares formula. By identifying it as a 2 − b 2 , we computed the individual squares and subtracted them. This technique is useful in various mathematical contexts.
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Recognize the expression as a difference of squares: ( a + b ) ( a − b ) where a = 10 and b = 2 8 .
Apply the difference of squares formula: ( a + b ) ( a − b ) = a 2 − b 2 .
Calculate the squares: ( 10 ) 2 = 10 and ( 2 8 ) 2 = 32 .
Subtract to find the final answer: 10 − 32 = − 22 .
Explanation
Understanding the Problem We are asked to multiply the expression ( 10 + 2 8 ) ( 10 − 2 8 ) . This expression is in the form of ( a + b ) ( a − b ) , which is a difference of squares.
Applying Difference of Squares Recall the difference of squares formula: ( a + b ) ( a − b ) = a 2 − b 2 . In our case, a = 10 and b = 2 8 .
Substitution Now, we substitute a and b into the formula: ( 10 + 2 8 ) ( 10 − 2 8 ) = ( 10 ) 2 − ( 2 8 ) 2
Calculating the Squares Next, we calculate the squares: ( 10 ) 2 = 10 ( 2 8 ) 2 = 2 2 × ( 8 ) 2 = 4 × 8 = 32
Final Calculation Now, we subtract the squares: 10 − 32 = − 22
Final Answer Therefore, ( 10 + 2 8 ) ( 10 − 2 8 ) = − 22 .
Examples
The difference of squares pattern is useful in various fields, such as engineering and physics, where simplifying complex expressions is crucial. For example, when calculating the energy difference between two states in quantum mechanics, you might encounter an expression in the form of ( a + b ) ( a − b ) . By recognizing this pattern, you can quickly simplify the expression to a 2 − b 2 , making the calculation more manageable. This technique is also used in signal processing to simplify filter designs and in financial modeling to analyze investment strategies.
