Problem 54 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given values of the variables. (Lesson 1.2 ) \(x^{2}+z\) when \(x=8\) and \(z=12\)

Step-by-Step Solution

Answer

The output of the expression \(x^{2}+z\) when \(x=8\) and \(z=12\) is \(76\)

Step 1: Substitute the Given Values

Replace the variables \(x\) and \(z\) in the equation \(x^{2}+z\) with their given values, \(x=8\) and \(z=12\) respectively. This gives us the new equation \(8^{2}+12\)

Step 2: Apply the Order of Operations

According to the order of operations (PEMDAS/BODMAS), we perform the exponent calculation before the addition. It results in \(64+12\)

Step 3: Perform the Addition

Add the two numbers together to get the final result, which is \(64 + 12 = 76\)

Key Concepts

Order of Operations

When evaluating expressions in algebra, it's crucial to apply the order of operations correctly. This order is commonly remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This ensures consistency and accuracy in solving mathematical problems. For instance, when given an expression such as \(x^{2}+z\), with \(x=8\) and \(z=12\), one must handle the exponent part \(x^{2}\) before moving on to the addition with \(z\).
In practice, begin by solving any operations within parentheses. Next, calculate exponents and roots. Only then should you proceed to multiplication and division, followed by addition and subtraction, consistently working from left to right. Misplacing these steps might lead to incorrect results. In educational terms, we often liken the correct application of the order of operations to following a recipe—it’s essential for the desired outcome.

Substitute Variables

Substituting variables is like replacing placeholders with their actual values. It's a vital part of solving algebraic expressions because it transforms abstract equations into concrete numerical calculations we can perform. To substitute variables effectively, first identify the variables and then replace them with the given or known values. In the expression \(x^{2}+z\), the variables are represented by \(x\) and \(z\), and they are 'placeholders' for the numbers you will be working with.
Once we substitute \(x\) with 8 and \(z\) with 12, the expression becomes \(8^{2} + 12\). This makes the equation ready for computation. It’s like personalizing the expression to fit the specific values provided. It is important to write down the substituted values clearly, to avoid any confusion as you proceed with the next steps in the calculation.

Exponent Calculation

Exponent calculation, often referred to as 'raising to a power', is an operation where a number, called the base, is multiplied by itself a certain number of times indicated by the exponent. In the expression \(8^{2}\), the base is 8, and the exponent is 2, meaning you need to multiply 8 by itself once, because the exponent indicates how many times the base is used as a factor.
Therefore, \(8^{2} = 8 \times 8 = 64\). Exponents are one of the operations that require careful attention because they can dramatically change the result of a calculation. Errors in exponent calculations are common and can stem from misunderstandings about how to treat the base and exponent, especially when they involve higher numbers or negative exponents. Always perform exponent calculations right after you’ve dealt with any parentheses, but before you do any multiplication, division, addition, or subtraction unless the exponent itself is within a parenthesis.