Problem 10 - College Algebra

Question

Evaluate each exponential expression. $$ -9^{0} $$

Step-by-Step Solution

Answer

Therefore, \(-9^{0} = 1\).

Step 1: Evaluate Exponential Expression

An exponential expression is an expression where a number, the base, is raised to a power, also called an exponent. In this expression, the base is -9 and the exponent is 0. Now, remember, any nonzero number to the power of 0 is 1. So, we simply apply this rule to our expression.

Step 2: Apply Exponent Rule

The rule states that any nonzero number \((-9\) in this case) raised to the 0 power equals 1. So, when we apply the rule to our expression, it simplifies to 1

Key Concepts

Exponent Rules

Understanding exponent rules is crucial when evaluating exponential expressions. An exponent tells us how many times to multiply the base number by itself. For example, when we see an expression like \( 3^{2} \), we multiply 3 by itself two times, which equals 9. There are several important rules to remember:

• The Product of Powers rule: To multiply two powers with the same base, add their exponents. For instance, \( a^{m} \times a^{n} = a^{m+n} \).
• The Quotient of Powers rule: To divide two powers with the same base, subtract the exponent of the denominator from the exponent of the numerator, as \( \frac{a^{m}}{a^{n}} = a^{m-n} \).
• The Power of a Power rule: To raise a power to another power, multiply the exponents, such as \( (a^{m})^{n} = a^{m \times n} \).
• The Zero Exponent rule, which applies to our textbook exercise, states that any non-zero base raised to the zero power is always 1: \( a^{0} = 1 \), where \( a \) is nonzero.

These rules help simplify complex expressions and solve them systematically.

Exponential Expressions

An exponential expression is composed of a base raised to an exponent. The base can be any real number, and the exponent can be positive, negative, or zero. The value of such expressions can grow very rapidly. For instance, expressions like \( 2^{10} \) may seem small, but they evaluate to 1024, a quite large number.
When dealing with exponential expressions, it's essential to evaluate them in the context of their rules. As seen with 'Exponent Rules', manipulating exponents properly can make calculations much easier. In our exercise, the exponential expression was \( -9^{0} \), which might appear unusual since the exponent is 0. However, applying the exponent rules leads to a simple solution.

Powers of Zero

The concept of 'powers of zero' refers to the straightforward rule that any nonzero number raised to the power of zero equals one. It is a standard rule in mathematics and can often cause confusion at first glance. Why does \( 7^{0} \) equal 1, not 0?
The logic behind this is based on the patterns exhibited by powers as they decrease. For example, as the exponent decreases, the value of the expression halves for base 2: \( 2^{3} = 8 \), \( 2^{2} = 4 \), \( 2^{1} = 2 \), and so on. When you reach zero, the pattern would imply you've divided by the base one final time, reaching 1. This rule applies across all nonzero bases, making it a universal simplifier in mathematics. In our textbook exercise, despite the base being a negative number, the rule is the same: \( -9^{0} = 1 \).

Simplifying Expressions

Simplifying expressions can transform a complex equation into something more manageable and easier to understand. This process often involves using exponent rules to condense the expression. It is a step-by-step procedure where we combine like terms, apply exponent rules, and carry out any arithmetic operations if necessary.
The goal is to write the expression in its simplest form. For instance, merging terms using the product of powers rule or reducing fractions using the quotient of powers rule. In the case of our exercise, the simplification step was direct and simple due to the exponent being zero. Remember, the more you practice these rules, the more intuitive simplifying expressions will become. It's like a math toolkit that you can use to tackle complex problems by breaking them down into simpler, more digestible pieces.