Problem 2 - Algebra 2 and Trigonometry

Question

Tony said that \(a^{0}+a^{0}=2 a^{0}=2 .\) Do you agree with Tony? Explain why or why not?

Step-by-Step Solution

Answer

Yes, Tony is correct; both sides equal 2, confirming his equation as valid.

Step 1: Understand the Definition of Exponents

Any non-zero number raised to the power of zero is equal to one. Therefore, if we have \(a^0\) where \(aeq0\), then \(a^0 = 1\).

Step 2: Evaluate the Given Expression

According to the expression presented by Tony, \(a^0 + a^0 = 2a^0\). Evaluate the left-hand side: \(a^0 + a^0 = 1 + 1 = 2\).

Step 3: Evaluate the Right-Hand Side

Now, look at the right-hand side of the equation, \(2a^0\). Since \(a^0 = 1\), then \(2a^0 = 2 \times 1 = 2\).

Step 4: Compare Both Sides

Both sides of the equation simplify to 2: \(a^0 + a^0 = 2\) and \(2a^0 = 2\). Since they are equal, Tony's statement holds true.

Key Concepts

Properties of Exponents

When diving into the world of exponents, one essential rule is the Zero Exponent Rule. This rule tells us that any non-zero number raised to the power of zero is equal to one. It's a crucial property among various **Properties of Exponents** and provides the foundation for understanding how different exponent rules interact. Here are a few key properties:

• **Zero Exponent**: Any non-zero number raised to the zero power results in one, expressed as \(a^0 = 1\) for \(a eq 0\).
• **Product of Powers**: When multiplying identical bases, you add the exponents, shown by \(a^m \times a^n = a^{m+n}\).
• **Quotient of Powers**: For division with the same base, subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\) for \(a eq 0\).
• **Power of a Power**: Raising an exponentiated number to another power involves multiplying the exponents \((a^m)^n = a^{m \times n}\).

Understanding these properties allows students to simplify and solve exponential expressions effectively, laying the groundwork for complex calculations and algebraic manipulations.

Exponentiation

Exponentiation is the mathematical operation where a number, known as the base, is raised to the power of an exponent. This is a crucial concept that underpins various algebraic structures and real-world applications. Here's how it works in practice:

• An exponent indicates how many times you multiply the base by itself. For instance, \(3^4 = 3 \times 3 \times 3 \times 3 = 81\).
• **Zero Exponent Rule** is particularly interesting in this operation. It states that any number except zero itself raised to the power of zero equals one \(a^0 = 1\).
• The operation can extend to negative exponents, transforming into reciprocal operations, such as \(a^{-n} = \frac{1}{a^n}\).

Exponentiation is essential for exponential growth calculations, scientific notation, and solving polynomial equations. Mastering how exponents work helps students grasp more sophisticated mathematical concepts with ease.

Equation Simplification

Simplifying equations involves reducing them to their most basic form, where like terms and consistent operations are streamlined. The simplification step not only clarifies mathematical operations but also removes any redundant complexity from expressions, making comparisons or further operations simpler. In terms of exponents, simplifying exponent-based equations plays out as:

• Combining like terms, as seen in the exercise where \(a^0 + a^0\) simplifies due to both terms equaling 1, thus summing to 2.
• Utilizing known properties, including the Zero Exponent Rule, ensures accurate reduction of terms.
• Breaking down expressions within parentheses, distributing coefficients, and using addition or subtraction consistently for consolidation.

In the exercise with Tony, through proper application of these principles, we confirm the equivalence showing detailed action on both sides of the equation: the left \(a^0 + a^0\) simplified to 2, matching the right \(2a^0\) and supporting Tony's claim. Understanding these simplification steps is crucial for conquering algebraic challenges efficiently.