Problem 56 - Beginning and Intermediate Algebra

Question

Evaluate each expression when \(x=1, y=3,\) and \(z=5.\) \(2 z^{2}\)

Step-by-Step Solution

Answer

The expression evaluates to 50.

Step 1: Understand the Expression

The expression to evaluate is \(2z^2\). This means we need to substitute the value of \(z\) into the expression and evaluate.

Step 2: Substitute the Given Value

We know that \(z = 5\). Substitute \(z = 5\) into the expression to get \(2(5)^2\).

Step 3: Calculate the Power

Find \((5)^2\), which is \(5 \times 5 = 25\).

Step 4: Multiply by the Coefficient

Multiply the result from Step 3 by 2: \(2 \times 25 = 50\).

Step 5: Final Result

The value of the expression \(2z^2\) when \(z = 5\) is 50.

Key Concepts

Substitution

Substitution is a fundamental technique in algebra that allows us to replace variables in an expression with specific values. This technique simplifies complex algebraic expressions and makes them easier to solve or evaluate. For example, in the expression \(2z^2\), if we are told that \(z = 5\), we can substitute \(5\) in place of \(z\).
This turns the expression into \(2(5)^2\). Applying substitution correctly is crucial because it transforms abstract expressions into numerical calculations that provide concrete answers. Using substitution saves time and reduces errors in solving algebraic problems.

Exponentiation

Exponentiation is a mathematical operation where a number, known as the base, is multiplied by itself a certain number of times as indicated by the exponent. In the expression \(z^2\), \(z\) is the base and the exponent is 2, indicating that you multiply \(z\) by itself once.

• For example, \((5)^2\) means \(5 \times 5\).
• This equals 25 because the exponent tells us to multiply the base by itself.

Understanding how to apply exponents is essential for evaluating expressions with powers. The procedure of exponentiation makes expressions more compact and manageable, especially when dealing with large numbers or complex algebraic terms.

Coefficients

Coefficients are numerical or constant factors in terms of an algebraic expression, typically placed in front of variables. In the expression \(2z^2\), \(2\) is the coefficient. Coefficients indicate how many times a term is considered in the equation.
Here, \(2\) means we take the result of \(z^2\) and multiply it by 2.

• Step 1 involved solving \(z^2\), which we found to be 25.
• Step 2 requires multiplying this 25 by the coefficient 2, resulting in 50.

Recognizing and properly applying coefficients in algebraic expressions helps in performing accurate calculations and understanding the weight each variable term holds in the expression.

Evaluation of Expressions

Evaluation of expressions refers to the process of calculating the numerical value of an algebraic expression by substituting variables with known values. This process combines various mathematical operations such as addition, subtraction, multiplication, division, and exponentiation.
In our example, evaluating \(2z^2\) involves several steps:

• First, substitute \(z\) with 5 to transform the expression into \(2(5)^2\).
• Then, calculate the exponentiation, \((5)^2 = 25\).
• Finally, multiply the result by the coefficient, giving us \(2 \times 25 = 50\).

Properly evaluating expressions is key in problem-solving, allowing us to turn abstract algebraic forms into specific numerical outcomes.