Problem 18 - Algebra A Combined Function

Question

Evaluate each expression with the given replacement values. $$ -4 x^{2} y^{3} \text { when } x=2 \text { and } y=-1 $$

Step-by-Step Solution

Answer

The evaluated expression is 16.

Step 1: Identify the Expression

The given expression is \(-4x^{2}y^{3}\). We need to evaluate this expression by substituting \(x = 2\) and \(y = -1\).

Step 2: Substitute Values into the Expression

Substitute \(x = 2\) and \(y = -1\) into the expression: \(-4(2)^{2}(-1)^{3}\).

Step 3: Calculate the Power of x

Calculate \(x^2\): \(2^2 = 4\). Now, the expression becomes \(-4 \times 4 \times (-1)^{3}\).

Step 4: Calculate the Power of y

Calculate \(y^3\): \((-1)^3 = -1\). Now, our expression is \(-4 \times 4 \times (-1)\).

Step 5: Perform Multiplication

Multiply the terms: \(-4 \times 4 = -16\) and then \(-16 \times (-1) = 16\).

Step 6: Final Calculation and Result

The evaluated expression is 16. Confirm all calculations are correct.

Key Concepts

Substitution Method

The substitution method is a crucial step when evaluating algebraic expressions. It involves replacing variables in the expression with specific values, which are usually provided in the problem statement. This method allows us to simplify the expression step by step, using known values instead of unknown variables.

• Identify each variable in the expression.
• Insert the given numerical values for these variables.

In our exercise, we start with the expression \(-4x^{2}y^{3}\). We substitute \(x = 2\) and \(y = -1\). After substitution, our expression becomes \(-4(2)^2(-1)^3\). This step sets the foundation for further calculations. By replacing variables with numbers, the expression turns into a series of multiplications, which are easier to solve.

Algebraic Operations

Algebraic operations include addition, subtraction, multiplication, and division, which are used to manipulate mathematical expressions. In algebra, understanding these operations is fundamental to solving problems.

• Ensure you follow the order of operations (PEMDAS/BODMAS): Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
• The Multiplication operation is applied after exponents are calculated.

For our expression \(-4 \times 4 \times (-1)\), we first handle the powers, and then we perform the multiplication step by step. Starting with the multiplication by \(-4\) and \(4\), which gives us \(-16\), and finally, multiplying the result by \(-1\) yields \(16\). This process illustrates the importance of carefully executing each operation to arrive at the correct answer.

Exponents in Algebra

Exponents are a mathematical notation indicating the number of times a number is multiplied by itself. In algebra, handling exponents is key to simplifying expressions. An exponent is written as a small number to the upper right of a base number.

• For example, in the term \(x^2\), \(x\) is the base and \(2\) is the exponent, meaning \(x\) is multiplied by itself once.
• Negative and zero exponents have special rules. For instance, \((-1)^3\) means multiplying \(-1\) by itself three times, resulting in \(-1\).

In solving the original exercise, we computed \(2^2 = 4\) for the variable \(x\) and \((-1)^3 = -1\) for \(y\). By understanding these basic exponent rules, calculating expressions involving exponents becomes a straightforward process. Always remember to handle exponents prior to multiplying through other terms.