Problem 10 - College Algebra

Question

Evaluate each exponential expression in Exercises 1–22. $$ -9^{0} $$

Step-by-Step Solution

Answer

The value of \(-9^0\) is 1.

Step 1: Understand the rule of exponents.

Any number (zero or nonzero), when raised to the power of 0, becomes 1, i.e., \(a^{0} = 1\), where \(a\) can be any number.

Step 2: Apply the rule to the given expression.

\(-9^{0}\) would equal to 1 because any number raised to the power of 0 is always 1.

Key Concepts

Rules of Exponents

Understanding the rules of exponents is essential for solving problems involving exponential expressions. These rules are like tools to simplify and manipulate powers, making it easier to work with complex algebraic expressions.

Here are some fundamental rules of exponents that students should know:

• The Product Rule: When multiplying two powers with the same base, you add their exponents, i.e., \( a^m \cdot a^n = a^{m+n} \).
• The Quotient Rule: When dividing two powers with the same base, you subtract the exponent of the denominator from the exponent of the numerator, i.e., \( \frac{a^m}{a^n} = a^{m-n} \).
• The Power of a Power Rule: When taking a power to another power, you multiply the exponents, i.e., \( (a^m)^n = a^{m \cdot n} \).
• The Power of a Product Rule: To find the power of a product, apply the exponent to each factor, i.e., \( (ab)^n = a^n \cdot b^n \).
• The Power of a Quotient Rule: To find the power of a quotient, apply the exponent to the numerator and denominator separately, i.e., \( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \) .

In applying these rules, it's important to pay attention to whether the base is negative, as this can affect the outcome when the exponent is even or odd.

Exponent of Zero

One of the most fundamental rules of exponents is understanding what happens when a number is raised to the power of zero. The rule states that any nonzero number raised to the zeroth power is always 1. Mathematically, this can be expressed as \( a^{0} = 1 \) for any nonzero \( a \).

The reasoning behind this rule lies in the properties of exponents. For example, using the quotient rule for exponents, if you divide any number by itself you get 1, and this can be illustrated as \( a^m / a^m = a^{m-m} = a^0 = 1 \).

This concept is crucial to grasp because it simplifies many exponential expressions and is frequently encountered across various areas of mathematics.

Evaluating Exponents

Evaluating exponential expressions requires understanding the specific 'power' that the base is being raised to. When we evaluate expressions like \( -9^{0} \), we are applying exponent rules to find out what this expression simplifies to.

As we learned with the exponent of zero, any number, unless it's zero itself, becomes 1 when raised to the zero power. Other important evaluations involve:

• Negative exponents, which denote reciprocals, i.e., \( a^{-n} = \frac{1}{a^n} \).
• Fractional exponents, where for instance \( a^{\frac{1}{n}} \) represents the nth root of a, and \( a^{\frac{m}{n}} \) is the nth root of \( a^m \).

Correctly evaluating exponents is fundamental in algebra and higher mathematics, laying the groundwork for solving equations and understanding functions involving exponents.

Algebraic Concepts

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. Exponents play a significant role in algebraic expressions and equations. Mastery of algebraic concepts allows students to solve a wide range of problems, from simple equations to more complex functions.

Key algebraic concepts involving exponents include:

• Distributive property, which allows us to expand expressions like \( a(b + c) \).
• Combining like terms, which means to sum or subtract terms that have the same variable parts.
• Solving equations, where exponents may require special techniques such as taking logarithms.
• Understanding polynomials, which often include terms with exponents and require operations such as factoring and expansion.

When working with exponents in algebra, it's also important to be mindful of the domain of the expression and whether the base can be zero or negative, as these scenarios might introduce additional considerations or restrictions.