If $x=3$ and $y=-1$, evaluate $2(x^2-y^3)$
Answer (2)
The value of the expression 2 ( x 2 − y 3 ) when x = 3 and y = − 1 evaluates to 20 . This is calculated by substituting the values into the expression, calculating the necessary powers, and simplifying the result. The final answer, after completing the operations, is 20 .
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Substitute x = 3 and y = − 1 into the expression 2 ( x 2 − y 3 ) .
Calculate x 2 = 3 2 = 9 and y 3 = ( − 1 ) 3 = − 1 .
Substitute the calculated values into the expression: 2 ( 9 − ( − 1 )) = 2 ( 9 + 1 ) = 2 ( 10 ) .
Simplify to find the final answer: 20 .
Explanation
Understanding the Problem We are given the expression 2 ( x 2 − y 3 ) and the values x = 3 and y = − 1 . Our goal is to substitute these values into the expression and evaluate it. This involves basic arithmetic operations such as squaring, cubing, and subtraction, followed by multiplication.
Substitution First, we substitute the given values of x and y into the expression: 2 ( x 2 − y 3 ) = 2 (( 3 ) 2 − ( − 1 ) 3 ) Now, we need to calculate 3 2 and ( − 1 ) 3 .
Calculating Exponents Next, we calculate the values of the exponents: 3 2 = 3 × 3 = 9 ( − 1 ) 3 = ( − 1 ) × ( − 1 ) × ( − 1 ) = − 1 Now we substitute these values back into the expression:
Simplifying the Expression Substituting the calculated values, we have: 2 ( 9 − ( − 1 )) Since subtracting a negative number is the same as adding its positive counterpart, we can rewrite the expression as: 2 ( 9 + 1 )
Final Calculation Now, we simplify the expression inside the parentheses: 9 + 1 = 10 So the expression becomes: 2 ( 10 ) Finally, we perform the multiplication: 2 × 10 = 20
Conclusion Therefore, the value of the expression 2 ( x 2 − y 3 ) when x = 3 and y = − 1 is 20.
Examples
Evaluating algebraic expressions is a fundamental skill in mathematics with applications in various fields. For instance, in physics, you might use such expressions to calculate the potential energy of an object given its position and other parameters. Similarly, in engineering, you could use algebraic expressions to model the behavior of circuits or structures. Understanding how to substitute values into expressions and simplify them is crucial for solving real-world problems in these areas.
