Evaluate $3 x^3-2 y^2$ when $x=-2$ and $y=4$
Answer (2)
To evaluate the expression 3 x 3 − 2 y 2 with x = − 2 and y = 4 , substitute the values, calculate each term, and combine them. The final result is − 56 . The computed value indicates the effect of the values substituted in the expression.
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Substitute x = − 2 and y = 4 into the expression 3 x 3 − 2 y 2 .
Calculate x 3 = ( − 2 ) 3 = − 8 .
Calculate y 2 = ( 4 ) 2 = 16 .
Evaluate the expression: 3 ( − 8 ) − 2 ( 16 ) = − 24 − 32 = − 56 . The final answer is − 56 .
Explanation
Understanding the Problem We are given the expression 3 x 3 − 2 y 2 and the values x = − 2 and y = 4 . Our goal is to substitute these values into the expression and evaluate it.
Substituting the Values First, we substitute the given values of x and y into the expression: 3 x 3 − 2 y 2 = 3 ( − 2 ) 3 − 2 ( 4 ) 2
Calculating x^3 term Next, we calculate ( − 2 ) 3 , which is ( − 2 ) × ( − 2 ) × ( − 2 ) = − 8 . So, we have: 3 ( − 2 ) 3 = 3 × ( − 8 ) = − 24
Calculating y^2 term Then, we calculate ( 4 ) 2 , which is 4 × 4 = 16 . So, we have: 2 ( 4 ) 2 = 2 × 16 = 32
Combining the Terms Now, we substitute these values back into the expression: 3 x 3 − 2 y 2 = − 24 − 32
Final Calculation Finally, we perform the subtraction: − 24 − 32 = − 56 Thus, the value of the expression 3 x 3 − 2 y 2 when x = − 2 and y = 4 is − 56 .
Examples
Imagine you are designing a fountain. The water flow can be modeled by the expression 3 x 3 − 2 y 2 , where x represents the water pressure and y represents the nozzle size. If you set the water pressure to -2 units and the nozzle size to 4 units, the resulting water flow would be -56 units. This calculation helps you understand how different settings affect the fountain's performance.
