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Chapter 5: Problem 1

Decide whether each expression is equal to \(0,1,\) or \(-1 .\) See Example 1. $$ 9^{0} $$

### Short Answer

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1

## Step by step solution

01

## Identify the base and exponent

The given expression is \(9^{0}\). Here, the base is 9 and the exponent is 0.

02

## Apply the zero exponent rule

According to the zero exponent rule, any non-zero number raised to the power of 0 is equal to 1. Mathematically, this is expressed as \(a^{0} = 1\) for any number \(a eq 0\).

03

## Apply the rule to the given expression

Since 9 is a non-zero number, \(9^{0} = 1\).

## Key Concepts

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

###### Zero Exponent Rule Explained

The zero exponent rule is one of the fundamental principles in the study of exponents. According to this mathematical rule, any non-zero number raised to the power of zero is always equal to 1. This can be represented as \(a^0 = 1\) where \(a\) is any non-zero number.

Let's break this down further. When we say \(9^0\), we are using the base 9 and raising it to an exponent of 0. By applying the zero exponent rule, we can directly conclude that \(9^0 = 1\). This is regardless of the base value when it is a non-zero number.

This concept simplifies many calculations, especially in algebraic expressions, where terms can be raised to the power of zero. Understanding and applying this rule can make problem-solving in algebra much more straightforward.

###### Understanding Exponents

Exponents, also known as powers or indices, represent how many times a number, known as the base, is multiplied by itself. For instance, \(9^3\) means 9 multiplied by itself three times: \(9 \times 9 \times 9 = 729\).

Exponents can make numbers either very large or very small. For example, \(10^6\) is a 1 followed by six zeros, which is 1,000,000. Conversely, negative exponents like \(10^{-3}\) can represent very small numbers: \(10^{-3} = 1/(10^3) = 0.001\).

When studying exponents, you will encounter several rules, including the zero exponent rule, product of powers rule, and power of a power rule. Each of these rules helps simplify complex algebraic expressions and makes calculations more manageable. Being comfortable with these rules is key to mastering algebra.

###### Algebraic Expressions and Mathematical Rules

Algebraic expressions are combinations of numbers, variables (letters that represent unknown values), and operations (such as addition, subtraction, multiplication, and division). An example is \(3x^2 + 5x - 7\). Here, \(3x^2\) is an algebraic term where 3 is the coefficient and \(x^2\) is the variable raised to an exponent.

Mathematics involves several rules and properties that guide us on how to manipulate these expressions. The commutative, associative, and distributive properties are some of these rules. Each property provides a systematic way of solving and simplifying expressions.

The zero exponent rule is a specific rule within this framework. It specifically addresses how to handle terms where variables or numbers are raised to zero. For instance, in an expression like \(4y^0 + 7\), knowing the zero exponent rule allows us to simplify it to \(4 \times 1 + 7 = 11\). Understanding these core concepts helps build a strong foundation in algebra and enhances problem-solving skills.

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