Problem 14 - Algebra 2 and Trigonometry

Question

In \(11-22,\) find the value of each expression when \(x \neq 0\) $$ -4^{0} $$

Step-by-Step Solution

Answer

The value of \(-4^0\) is -1.

Step 1: Understand the Expression

The expression given is \(-4^0\).. Here, the base is \(-4\) and it is raised to the power of \(0\).

Step 2: Apply the Exponent Rule

According to the exponent rule, any non-zero number raised to the power of 0 is equal to 1. Thus, \((-4)^0 = 1\).

Step 3: Consider Negative Sign Placement

Notice that the negative sign is not inside the parentheses. Hence, it is not part of the base being raised to the 0; therefore, we calculate \(| -1 \times 4^0 |\).

Step 4: Evaluate the Correct Expression

We evaluate \(4^0\) first, which equals \(1\). Thus, the expression becomes \(-1 imes 1 = -1\)..

Key Concepts

Exponent Rules

Exponents are a shorthand way to represent repeated multiplication. When we talk about exponent rules, we are referring to the guidelines that help us simplify expressions involving exponents. Let's go through some of the most important aspects:

• Zero Exponent Rule: Any non-zero base raised to the power of zero is always equal to one. This rule stems from the pattern you observe when you repeatedly divide by the base, scaling down the power.
• Product of Powers Property: When multiplying two expressions with the same base, you add the exponents, \(a^m \times a^n = a^{m+n}\).
• Power of a Power Property: When raising an exponent to another power, multiply the exponents, \( (a^m)^n = a^{m \cdot n} \).
• Power of a Product Property: Each factor in a product can be raised to the power separately, \( (ab)^n = a^n \cdot b^n \).

Understanding these rules will allow you to confidently simplify or evaluate expressions with exponents.

Negative Bases

Negative bases can be tricky, especially when it comes to evaluating expressions with exponents. Here are some guidelines:

• Even vs. Odd Powers: When a negative number is raised to an even power, the result is positive. For example, \( (-2)^2 = 4 \). However, if a negative number is raised to an odd power, the result is negative, like \( (-2)^3 = -8 \).
• Without Parentheses: Pay attention to parentheses! If a negative sign is outside the base and the base is raised to an exponent, the negative remains. For example, \( -4^2 \) is interpreted as \( -(4^2) = -16 \).
• With Parentheses: When a base with a negative sign is put within parentheses and raised to a power, the exponent applies to the entire base. For instance, \( (-3)^3 = -27 \).

Remembering the role of parentheses will help determine whether the negative sign is part of the base or not.

Order of Operations

The order of operations is a set of rules that dictates the sequence in which different operations must be performed within a mathematical expression. The common acronym used to recall the order of operations is PEMDAS:

• Parentheses: Always resolve expressions within parentheses first.
• Exponents: Consider exponents next, so that any power is calculated before moving on.
• Multiplication and Division: Perform these operations as they appear from left to right. They have equal precedence, meaning neither is performed before the other beyond their sequence in the expression.
• Addition and Subtraction: Finally, tackle both these operations, also working from left to right. Like multiplication and division, they share equal precedence.

In the given exercise, considering the order of operations ensures that we calculate the exponent first (\(4^0 = 1\)), and then apply the negative sign, leading us to correctly evaluate \(-1 \cdot 1 = -1\). Following this sequence prevents mathematical confusion and yields correct results.