Problem 62 - Algebra 1: Concepts and Skills

Question

Give a reason for each step to show that the definitions of zero and negative exponents hold true for the properties of exponents. $$ \begin{aligned} &a^{0}=a^{n-n}\\\ &=\frac{a^{n}}{a^{n}}\\\ &=1 \end{aligned} $$

Step-by-Step Solution

Answer

The exercise demonstrates that a number raised to the power of zero equals one, in accordance with the rules of exponents.

Step 1: Apply definition of a zero exponent

The expression \(a^{0}\) can be rewritten as \(a^{n-n}\), that is because any number subtracted by itself equals zero. Therefore, \(n - n\) results in zero, giving us \(a^{0}\) or \(a^{n-n}\).

Step 2: Apply rules of exponents

According to the rules of exponents, \(a^{n-n}\) can be rewritten as \(\frac{a^{n}}{a^{n}}\). That is because subtracting exponents is equivalent to dividing the base raised to these exponents.

Step 3: Simplify expression

The expression \(\frac{a^{n}}{a^{n}}\) is equal to 1. That is because any number divided by itself equals 1, regardless of what the base (\(a\)) or the exponent (\(n\)) is.

Key Concepts

Understanding the Zero Exponent

Have you ever wondered why any non-zero number to the power of zero equals one? Let's unravel this concept. When we talk about the zero exponent, we're exploring a fundamental property of exponents: the expression \(a^0 = 1\) for any non-zero number \(a\). This might seem puzzling at first, but it makes perfect sense when you break it down.

Consider the expression \(a^{n-n}\). Here, \(n - n = 0\) simplifies the exponent to zero, producing \(a^0\). Now, applying the properties of exponents, specifically division, we see \(a^{n-n}\) as \(\frac{a^n}{a^n}\). Since any number divided by itself equals one, \(\frac{a^n}{a^n} = 1\). Thus, no matter what number \(a\) might be (except zero), when raised to the zero power, it is always 1. This is a crucial point of understanding in exponent algebra, as it helps maintain consistency in mathematical operations.

Exploring Negative Exponents

Negative exponents can often be misunderstood, but they are easier to grasp once you know the rule: a negative exponent indicates the reciprocal of the base raised to the opposite positive exponent. To put it simply, if you have \(a^{-n}\), it translates to \(\frac{1}{a^n}\).

Let's consider an example of \(a^{-3}\). By the definition of negative exponents, this becomes \(\frac{1}{a^3}\). It essentially flips the base \(a\) from the numerator to the denominator, presenting the reciprocal.

When dealing with negative exponents, it's important to remember these key points:

• The base must be non-zero.
• The larger the magnitude of the negative exponent, the smaller the resulting value is, since you're taking powers of fractions.
• Grasping negative exponents expands your capability to simplify and solve complex algebraic expressions efficiently.

The Rules of Exponents

The rules of exponents are a set of guidelines that help simplify expressions involving exponents. Let's highlight some of the key rules that apply to powers.

• **Product Rule**: When you multiply two expressions with the same base, you add their exponents: \(a^m \times a^n = a^{m+n}\).
• **Quotient Rule**: Dividing two expressions with the same base involves subtracting the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
• **Power Rule**: When you raise a power to another power, you multiply the exponents: \((a^m)^n = a^{m\times n}\).
• **Negative Exponents**: This indicates taking the reciprocal of the base: \(a^{-n} = \frac{1}{a^n}\).
• **Zero Exponent**: Any non-zero base raised to the zero power is always one: \(a^0 = 1\).

Each of these rules provides a systematic way to deal with situations where mathematics might otherwise seem confusing. Properly applying these rules ensures accurate simplification of exponential expressions, paving the way for solving complicated problems with ease. Keep practicing these rules, and over time, you'll find handling exponents becomes second nature.