Problem 4 - A Graphical Approach to College Algebra

Question

Simplify each expression. Leave answers with exponents. $$-2^{0}$$

Step-by-Step Solution

Answer

The expression simplifies to -1.

Step 1: Recognize the Exponent Rule

Recall that any number raised to the power of zero is 1. This mean \( a^0 = 1 \) for any \( a\) that is not zero.

Step 2: Apply the Zero Exponent Rule

Apply the zero exponent rule to the expression: \( 2^0 \) which simplifies to 1 since \( 2^0 = 1 \).

Step 3: Consider the Negative Sign

The expression is \(-2^0\). In this context, the negative sign is outside the power function. Thus, we have \(- (2^0) = -1\).

Key Concepts

Zero Exponent Rule

The Zero Exponent Rule is a fundamental concept in mathematics. It states that any non-zero number raised to the power of zero equals one. This might sound peculiar at first, but there's a logical explanation behind it.
This rule is consistent because extending the pattern of exponential division leads to this outcome. For example, when dividing powers of the same base:

• \( a^n \div a^n = a^{n-n} = a^0 \)
• Since any number divided by itself equals one, \( a^0 = 1 \)

It's crucial to remember that this rule only applies to bases not equal to zero, as zero raised to the zero power is generally considered indeterminate. By understanding and applying the zero exponent rule properly, you can simplify many complex expressions with ease.

Negative Sign in Exponents

Handling a negative sign in expressions with exponents requires extra attention. In the expression \(-2^0\), the negative sign is applied to the whole term after evaluating it with the exponent rule.
Consider this breakdown:

• The number part, \(2^0\), simplifies to \(1\) due to the zero exponent rule.
• The negative sign is not part of the base; it's separate and applied afterward.
• Thus, we rewrite it as \(- (2^0)\), which simplifies to \(-1\).

This process distinguishes between cases where the negative might be inside the power, affecting base calculation, and when it's outside, like here, only altering the final result's sign.

Simplification Steps

Simplifying expressions with exponents involves a few clear steps. Always ensure to address the exponent before applying other operations like addition or subtraction.
Let's follow our example:

• **Identify the Rule**: Recognize applicable exponent rules. For \(2^0\), use the zero exponent rule, simplifying it to \(1\).
• **Address the Negative**: A negative sign outside the exponent simply negates the entire result post simplification. Apply it after handling the exponent, leading to \(-1\) for our example.

By repeatedly practicing these steps, tackling more complex expressions can become much more straightforward.
Always double-check to see each part of the expression is correctly evaluated according to these guidelines.