Problem 43 - College Algebra with Corequisite Support

Question

For the following exercises, simplify the expression. \(7 z-3+z \times 6^{2}\)

Step-by-Step Solution

Answer

The simplified expression is \(43z - 3\).

Step 1: Identify Terms

Look at the expression: \(7z - 3 + z \times 6^2\). It consists of two separate parts to consider: the term \(7z - 3\) and the multiplication term \(z \times 6^2\).

Step 2: Calculate Exponents

Calculate the exponent in the expression. Here, \(6^2\) needs to be simplified first.\[6^2 = 36\]

Step 3: Apply Multiplication

Using the result from Step 2, multiply the expression by \(z\):\[z \times 36 = 36z\]

Step 4: Combine Like Terms

The expression now is \(7z - 3 + 36z\). Combine the like terms, \(7z + 36z\):\[43z - 3\]

Step 5: Write Simplified Expression

The fully simplified expression from combining like terms is \[43z - 3\].

Key Concepts

Combining Like Terms

In algebra, simplifying expressions often involves combining like terms to make calculations simpler and clearer. Like terms are those that have the exact same variable raised to the same power. This means that the variable part of the terms must match completely, even if the coefficients (the numbers in front of the variables) are different.

In our example, after dealing with exponents and multiplication, we arrived at the expression:

• Terms: \(7z - 3 + 36z\)

Notice that \(7z\) and \(36z\) are like terms because they both contain the variable \(z\) raised to the same power (which is 1). So, to combine them, you simply add or subtract their coefficients:

• \(7z + 36z = 43z\)

The constant term \(-3\) does not have a matching term and stays separate. The combined expression becomes \(43z - 3\), which is the simplified form of the original expression using combining like terms.

Exponents

Understanding exponents is crucial because they indicate repeated multiplication of a base number. In the expression \(z \times 6^2\), the number 6 is raised to the power of 2, written as \(6^2\). This means 6 is multiplied by itself:

• \(6 \times 6 = 36\)

Calculating the exponent simplifies the portion of the expression where it appears. Exponents make it efficient to express and calculate large numbers, as seen here. This simplification makes handling more complex algebraic expressions much more straightforward since it reduces multi-step operations into simpler components.

Multiplication in Algebra

Multiplication in algebra is used to expand expressions or to combine numbers with powers and variables. In our worked example, once the exponent was resolved to \(36\), the next step was involving multiplication:

• Expression: \(z \times 36\)

Here, \(z\) is multiplied by 36. Multiplication of a variable by a number involves distributing the number across the variable, which results in:

• \(z \times 36 = 36z\)

This step is crucial for simplifying an algebraic expression, as it ensures all parts of the expression are accounted for and fully integrated when combining terms later. The result of this simplification, \(36z\), is then used in the next phase, which involves combining like terms to achieve the final simplified expression.