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Evaluate the algebraic expression $x^2 + y^3 - z$ with x=7, y=-8, z=7
Mathematics
College Readiness (High School Senior/College Freshman)
Question Content
Evaluate $x^2 + y^3 - z$ for $x = 7$, $y = -8$, and $z = 7$.
Correct Answer
-448
Detailed Solution Steps
1
Step 1: Substitute the given values into the expression. Replace $x$ with 7, $y$ with -8, and $z$ with 7, resulting in: $7^2 + (-8)^3 - 7$
2
Step 2: Calculate each term separately. First, $7^2 = 7 \\times 7 = 49$. Next, $(-8)^3 = (-8) \\times (-8) \\times (-8) = -512$. The final term is just 7.
3
Step 3: Combine the calculated terms using the given operations: $49 + (-512) - 7$
4
Step 4: Simplify the expression. First, $49 - 512 = -463$, then $-463 - 7 = -448$.
Knowledge Points Involved
1
Substitution into algebraic expressions
This is the process of replacing variables in an expression with their given numerical values to evaluate the expression. It is a foundational skill for solving algebraic problems, used whenever you need to find the numerical value of an expression with known variable values.
2
Exponentiation of positive and negative numbers
Exponentiation means multiplying a number by itself the number of times indicated by the exponent. For positive bases, $a^n$ is positive for any positive integer $n$. For negative bases, if the exponent is odd, the result is negative (e.g., $(-8)^3 = -512$); if the exponent is even, the result is positive.
3
Integer arithmetic with positive and negative numbers
This involves adding, subtracting, multiplying, and dividing positive and negative integers. Key rules include: adding a negative number is equivalent to subtracting its positive value, and subtracting a number is equivalent to adding its negative inverse.
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