Problem 20 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the expression. Write fractional answers in simplest form.\((-2)^{0}\)

Step-by-Step Solution

Answer

The result of \((-2)^{0}\) is 1.

Step 1: Expression

Evaluate the expression. Write fractional answers in simplest form.\((-2)^{0}\)

Step 2: Exponent Rules

\(a^0=1\), \((a^m)^n=a^{mn}\), \(a^m a^n=a^{m+n}\).

Step 3: Result

The result of \((-2)^{0}\) is 1.

Key Concepts

Exponents in Algebra

When you encounter exponents in algebra, you're dealing with a shorthand method of expressing repeated multiplication. An exponent tells you how many times to multiply a base number by itself. A basic form of an exponential expression is \( b^n \), where \( b \) is the base and \( n \) is the exponent.For instance, \( 3^4 \) indicates that you should multiply 3 by itself 4 times: \( 3 \times 3 \times 3 \times 3 \). When learning about exponents, it's also crucial to know about special cases such as the zero exponent rule, which states that any nonzero number raised to the power of zero is one \( (b^0 = 1) \). This rule is a fundamental concept in algebra that simplifies many expressions and serves as a stepping stone to more complex operations such as exponential growth and decay.

Simplifying Expressions

The goal of simplifying expressions is to rewrite an algebraic expression in the simplest form possible. When working with exponents, simplification often involves applying exponent rules to make expressions more manageable. For example, the expression \( x^a \cdot x^b \) can be simplified to \( x^{a+b} \) by using the product of powers property, which tells us to add exponents when multiplying like bases.Simplifications are beneficial because they can make otherwise complex calculations easy to understand. As seen in the zero exponent example \( (-2)^0 = 1 \), simplifying allows us to evaluate algebraic expressions quickly and clearly, without the need for extensive computations. Remember, the more you are comfortable with these rules, the better you will get at identifying simplification opportunities in algebraic expressions.

Properties of Exponents

Understanding the properties of exponents is essential in algebra. These properties include the product of powers, quotient of powers, power of a power, and, as highlighted by our example, the zero exponent rule. By mastering these properties, you will be able to handle a wide range of algebraic tasks.

• Product of Powers: Use this property when you multiply like bases. Add the exponents.
• Quotient of Powers: Use this property when you divide like bases. Subtract the exponents.
• Power of a Power: When a power is raised to another power, multiply the exponents.
• Zero Exponent Rule: As seen with \( (-2)^0 = 1 \), any nonzero base raised to the power of zero equals one.
• Negative Exponent Rule: \( b^{-n} \) equates to \(\frac{1}{b^n}\) and means that you should take the reciprocal of the base raised to the positive exponent.

By recognizing these properties, you can simplify expressions quickly and correctly. This knowledge serves as a key to unlocking the power of algebra and is extensively used when dealing with more advanced mathematics, such as calculus and differential equations.