Find the value of the expression: [tex]\frac{1}{21} x^2 y^2 \text { for } x=-\frac{1}{3}, y=4 \frac{1}{2}[/tex]
Answer (2)
Substitute the given values of x = − 3 1 and y = 2 9 into the expression 21 1 x 2 y 2 .
Calculate x 2 = ( − 3 1 ) 2 = 9 1 and y 2 = ( 2 9 ) 2 = 4 81 .
Multiply the results: 21 1 ⋅ 9 1 ⋅ 4 81 = 756 81 .
Simplify the fraction: 756 81 = 28 3 . The final answer is 28 3 .
Explanation
Understanding the Problem We are given the expression 21 1 x 2 y 2 and the values x = − 3 1 and y = 4 2 1 = 2 9 . Our goal is to substitute these values into the expression and simplify to find the final value.
Substituting the Values First, we substitute the given values of x and y into the expression: 21 1 x 2 y 2 = 21 1 ( − 3 1 ) 2 ( 2 9 ) 2
Calculating the Squares Next, we calculate the squares of x and y : x 2 = ( − 3 1 ) 2 = 9 1 y 2 = ( 2 9 ) 2 = 4 81
Substituting Back into the Expression Now, we substitute these values back into the expression: 21 1 x 2 y 2 = 21 1 ⋅ 9 1 ⋅ 4 81
Multiplying the Fractions We can simplify the expression by multiplying the fractions: 21 1 ⋅ 9 1 ⋅ 4 81 = 21 ⋅ 9 ⋅ 4 81 = 756 81
Simplifying the Fraction Finally, we simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 27: 756 81 = 756 ÷ 27 81 ÷ 27 = 28 3
Final Answer Therefore, the value of the expression is 28 3 .
Examples
Imagine you are calculating the area of a shape that depends on two variables, x and y . The formula for the area is given by A = 21 1 x 2 y 2 . If you know the values of x and y , you can substitute them into the formula to find the area. For example, if x = − 3 1 meters and y = 4 2 1 meters, then the area would be 28 3 square meters. This type of calculation is useful in various fields such as engineering, physics, and architecture, where you need to determine the size or scale of objects based on given parameters.
The expression 21 1 x 2 y 2 evaluates to 28 3 when substituting x = − 3 1 and y = 4 2 1 (or 2 9 ). This is achieved through squaring the values, multiplying the fractions, and simplifying the result. Final answer: 28 3 .
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