What is the following product?
$(5 \sqrt{2}-4 \sqrt{3})(5 \sqrt{2}-4 \sqrt{3})$
Answer (2)
The product ( 5 2 − 4 3 ) 2 equals 98 − 40 6 . This is achieved using the binomial square formula and simplifying the expression. Therefore, the final answer is 98 − 40 6 .
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Recognize the expression as a square of a binomial: ( 5 2 − 4 3 ) 2 .
Apply the binomial square formula: ( a − b ) 2 = a 2 − 2 ab + b 2 .
Calculate each term: a 2 = 50 , b 2 = 48 , and 2 ab = 40 6 .
Simplify the expression to obtain the final answer: 98 − 40 6 .
Explanation
Understanding the problem We are asked to find the product of (5
\[ \sqrt{2}-4 \sqrt{3})(5 \sqrt{2}-4 \sqrt{3}) \]
. This is equivalent to squaring the binomial (5 \sqrt{2}-4 \sqrt{3})$.
Applying the binomial square formula To find the product, we can use the formula ( a − b ) 2 = a 2 − 2 ab + b 2 , where a = 5 2 and b = 4 3 .
Calculating a^2 First, we calculate a 2 = ( 5 2 ) 2 = 5 2 × ( 2 ) 2 = 25 × 2 = 50 .
Calculating b^2 Next, we calculate b 2 = ( 4 3 ) 2 = 4 2 × ( 3 ) 2 = 16 × 3 = 48 .
Calculating 2ab Now, we calculate 2 ab = 2 ( 5 2 ) ( 4 3 ) = 2 × 5 × 4 × 2 × 3 = 40 6 .
Substituting the values Substituting these values into the formula, we get ( 5 2 − 4 3 ) 2 = 50 − 40 6 + 48 .
Simplifying the expression Finally, we simplify the expression: 50 + 48 − 40 6 = 98 − 40 6 .
Final Answer Therefore, the product ( 5 2 − 4 3 ) ( 5 2 − 4 3 ) is equal to 98 − 40 6 .
Examples
Imagine you are designing a garden and need to calculate the area of a square plot where the side length is expressed in terms of square roots. This type of calculation is useful in various fields, such as engineering, physics, and architecture, where precise measurements involving irrational numbers are required. For instance, if the side of the square is ( 5 2 − 4 3 ) , then the area would be ( 5 2 − 4 3 ) 2 = 98 − 40 6 . This calculation helps in determining the exact amount of materials needed or the space occupied.
