Problem 43 - College Algebra

Question

For the following exercises, simplify the expression. $$ 7 z-3+z \cdot 6^{2} $$

Step-by-Step Solution

Answer

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

Step 1: Calculate the Exponent

First, resolve the exponent in the expression. Evaluate \(6^{2}\). This is equal to \(36\). So, the expression now becomes \(7z - 3 + z \cdot 36\).

Step 2: Distribute

Next, multiply \(z\) by \(36\). The expression \(z \cdot 36\) becomes \(36z\). Now, rewrite the expression as \(7z - 3 + 36z\).

Step 3: Combine Like Terms

Combine the like terms \(7z\) and \(36z\). Adding these gives you \(43z\). Therefore, the expression simplifies to \(43z - 3\).

Key Concepts

Simplifying Expressions

Simplifying algebraic expressions involves reducing the expression to its simplest form so that it's easier to work with, either for solving or further analysis.
To simplify, we follow a sequence of mathematical operations, respecting the order of operations rules, commonly known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

• Start by resolving any exponents. Exponents are a mathematical shorthand for repeated multiplication, which simplifies expressions significantly by removing multiple multiplication signs.
• Perform multiplication and division from left to right. This step often includes distributing numbers across terms in parentheses.
• Finally, perform addition or subtraction from left to right, combining like terms wherever possible to simplify further.

This structured approach ensures clarity and accuracy, ultimately breaking down complex expressions into manageable parts.

Exponents

Exponents provide a convenient way to represent repeated multiplication. For example, the expression \(6^2\) implies multiplying 6 by itself, resulting in 36.
Understanding exponents is crucial for both arithmetic operations and algebraic simplification. Here are some key points about exponents:

• An exponent of 2 signifies squaring a number, meaning the base is multiplied by itself once.
• Exponents follow specific rules when multiplied or divided, such as \(a^m \cdot a^n = a^{m+n}\) or \(\frac{a^m}{a^n} = a^{m-n}\).
• Handling negative exponents involves taking the reciprocal of the base, like \(a^{-n} = \frac{1}{a^n}\).
• Any number raised to the power of zero equals one, known as the zero exponent rule, \(a^0 = 1\).

In our exercise, we started by handling \(6^2\) to simplify the expression right away.

Combining Like Terms

Combining like terms is a fundamental algebraic process used to simplify expressions by bringing similar terms together.
This means adding or subtracting terms that have the same variable part or "like" factors.

• The coefficient (the numerical part) is what you add, while the variables and exponents remain unchanged unless otherwise simplified.
• In the expression \(7z + 36z\), both terms include the variable \(z\), making them "like terms." So, you add only the coefficients: \(7 + 36 = 43\), giving you \(43z\).
• This step makes complex expressions more manageable and reduces errors in computations by minimizing the number of terms involved.
• Always ensure the terms you combine share identical variables and exponents, otherwise they cannot be combined.

Through this process, we simplified the initial expression to a more workable form, \(43z - 3\), by merging \(7z\) and \(36z\).