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Chapter 0: Problem 105

Explain the difference between the expressions \(6 x^{0}\) versus \((6 x)^{0}\).

Short Answer

Expert verified
The expression \(6 x^0\) equals 6, while \((6 x)^0\) equals 1.

Step by step solution

01

Understand Exponents

Recall that any nonzero number raised to the power of 0 is equal to 1. This is a fundamental rule in exponents. Mathematically, this can be written as follows: \[ a^0 = 1 \] where \(a eq 0\).
02

Evaluate \(6 x^0\)

The expression \(6 x^0\) can be evaluated by applying the exponent rule. The term \(x^0\) is equal to 1, so the expression simplifies to: \[ 6 x^0 = 6 \cdot 1 = 6 \]
03

Evaluate \((6x)^0\)

The expression \((6x)^0\) treats \(6x\) as a single term raised to the power of 0. According to the exponent rule: \[ (6x)^0 = 1 \]
04

Compare the Results

Notice that \(6 x^0\) simplifies to 6, whereas \((6 x)^0\) simplifies to 1. They demonstrate different outcomes based on how the exponent is applied.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Zero Exponent Rule
The zero exponent rule is a key concept in algebra. It states that any nonzero number raised to the power of zero is equal to 1. This can be written as \ a^0 = 1 \ where \( a eq 0 \). This rule is consistent because we can think of it as an extension of the properties of exponents. For instance, dividing two powers with the same base: \ \frac{a^1}{a^1} = a^{1-1} = a^0 \ Since \( a^1 \) divided by \( a^1 \) is 1, it follows that \( a^0 = 1 \). Remember, this applies to any nonzero base, whether it's a number, a variable, or a combination of both.
Evaluating Expressions
When evaluating algebraic expressions, understanding each term and operation is essential. Let's look at two expressions: \( 6 x^0 \) and \( (6x)^0 \). For \( 6 x^0 \), apply the zero exponent rule to \( x^0 \) first, which gives 1: \ 6 x^0 = 6 \cdot 1 = 6 \ Here, only \( x \) is raised to the power of zero, not 6. For \( (6x)^0 \), treat the entire term \( 6x \) as a single unit: \ (6x)^0 = 1 \ According to the zero exponent rule, any nonzero number or expression raised to the power of zero is 1.
Simplifying Algebraic Expressions
When simplifying algebraic expressions, the key is to follow the rules and perform operations step by step. This includes applying exponent rules correctly. Let's break down the expressions: - **Simplifying \( 6 x^0 \):** \( x^0 \) simplifies to 1 because of the zero exponent rule. Multiply 6 by 1 to get 6. - **Simplifying \( (6x)^0 \):** The entire expression \( 6x \) is considered together; apply the zero exponent rule directly, resulting in 1. Compare the two expressions: \( 6 x^0 \) results in 6, while \( (6x)^0 \) simplifies directly to 1. This demonstrates the importance of how exponents are applied in terms of grouping and placement.

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