Problem 6 - A Graphical Approach to College Algebra

Question

Simplify each expression. Leave answers with exponents. $$(-4 z)^{0}, z \neq 0$$

Step-by-Step Solution

Answer

The simplified expression is 1.

Step 1: Apply Zero Exponent Rule

Recall that for any non-zero number or variable, raising it to the power of zero equals one. This means for any expression \[ a^0 = 1 \], where \( a eq 0 \). Since \( z eq 0 \), this rule can be applied directly to the expression \((-4z)^0\) so that \((-4z)^0 = 1 \).

Step 2: Simplify

Based on the zero exponent rule from step 1, the expression \((-4z)^0\) simplifies directly to \(1\). There's no further simplification needed.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving numbers or expressions termed as the "base" and "exponent". It is used to represent repeated multiplication. For example, in the expression \( a^n \), "\( a \)" is the base, and "\( n \)" is the exponent. This expression means that you multiply "\( a \)" by itself "\( n \)" times.
Key points to remember include:

• A positive whole number as an exponent tells you how many times to multiply the base by itself.
• When the exponent is zero, any non-zero base raised to the power of zero equals 1, known as the Zero Exponent Rule.

This Zero Exponent Rule is particularly useful in algebra. It simplifies expressions and helps in solving equations quickly. Hence, simplifying expressions like \((-4z)^0\) to 1 becomes straightforward with this rule.

Simplification

Simplifying expressions in algebra often involves reducing them to their simplest form. It's about making expressions easier to work with without changing their values. In the context of the exercise, we use the Zero Exponent Rule to simplify \((-4z)^0\) to 1.

Steps to simplify using zero or other exponent rules usually involve:

• Identifying the base and the exponent in the expression.
• Applying known exponent rules such as Zero Exponent Rule, Product of Powers, and Power of a Power Rule.
• Reducing the expression to its simplest form.

The simplification process reduces complexity and helps in solving mathematical problems efficiently. Students often encounter simplifications involving rules of exponents in algebra and calculus.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They allow us to represent mathematical ideas in a general way. For example, \(-4z\) is an algebraic expression with "-4" as a coefficient and "z" as a variable.

In algebra:

• Variables represent numbers that can change.
• Constants are fixed values.
• Coefficients are numbers multiplying the variables.

Expressions can also include exponents, which add another layer of abstraction and functionality. Sometimes, expressions have to be simplified by applying rules like the Zero Exponent Rule as shown in the exercise. The process of working with these expressions is a critical skill in algebraic problem-solving, forming a base for more advanced math topics.