Problem 3 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. $$ 6^{0} $$

Step-by-Step Solution

Answer

The value of the expression \( 6^{0} \) is 1.

Step 1: Identify the exponent rule

The first step in solving this problem is to identify the relevant rule of exponents. In this case, the rule being applied is that any number (except 0) raised to the power of 0 equals 1.

Step 2: Apply the rule

Applying this rule to the given expression, \( 6^{0} \), means that we replace the entire expression with the number 1.

Key Concepts

Exponent Rules

Exponents describe how many times to multiply a number by itself, which is a crucial concept in algebra. To manipulate them correctly, one must understand the exponent rules, which are guidelines for performing operations involving exponents. Here are some key exponent rules:

• Product of Powers Rule: When multiplying two numbers with the same base, you add the exponents. For example, \(a^m \times a^n = a^{m+n}\).
• Quotient of Powers Rule: When dividing two numbers with the same base, you subtract the exponents. This can be represented as \(\frac{a^m}{a^n} = a^{m-n}\).
• Power of a Power Rule: When raising a power to another power, you multiply the exponents, represented by \((a^m)^n = a^{m \times n}\).
• Power of a Product Rule: When raising a product of numbers to a power, you can apply the exponent to each factor individually, as in \((ab)^n = a^n \times b^n\).
• Power of a Quotient Rule: Similar to the product rule, it involves distributing the exponent to both the numerator and denominator: \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\).

Knowing these rules helps simplify expressions and solve exponential equations efficiently.

Zero Exponent

The zero exponent rule states that any non-zero number raised to the power of zero equals one. This might seem counterintuitive at first, but it emerges logically from the rules of exponents. To understand why, consider how dividing powers of the same base leads to this rule.For example, using the quotient of powers rule, \(\frac{a^m}{a^m} = a^{m-m} = a^0\). Since any number divided by itself is one, we find that \(a^0 = 1\).

This rule is helpful because it simplifies expressions without needing further computation. It’s important to note that the base cannot be zero when applying this rule because \(0^0\) is undefined. Applying the zero exponent rule in math problems effectively can save time and reduce complexity.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. They are foundational in algebra and appear in numerous mathematical contexts, including solving equations and modeling real-world problems.The core components of algebraic expressions include:

• Constants: These are fixed numerical values.
• Variables: Symbols, often letters, that represent numbers and can change or vary.
• Coefficients: Numbers that multiply the variables. For instance, in \(3x\), 3 is the coefficient.
• Terms: Parts of the expression separated by plus or minus signs; for example, \(5x + 3\) has two terms, \(5x\) and \(3\).

When dealing with algebraic expressions, understanding the role of each component is crucial. Simplifying expressions using exponent rules, such as the zero exponent rule, can greatly assist in making the algebraic work more manageable.