Problem 11 - Intermediate Algebra

Question

Simplify each numerical expression. \(\left(-\frac{3}{4}\right)^{0}\)

Step-by-Step Solution

Answer

The expression \((-\frac{3}{4})^0\) simplifies to 1.

Step 1: Understand the Zero Exponent Rule

According to the zero exponent rule, any non-zero number raised to the power of zero equals one. This means that no matter what the base is (as long as it's not zero), if the exponent is zero, the result is always one.

Step 2: Apply the Zero Exponent Rule

Given the expression \( \left(-\frac{3}{4}\right)^{0} \), we apply the zero exponent rule. Here, the base is \(-\frac{3}{4}\), which is a non-zero number. Therefore, raising it to the power of zero gives us \(1\).

Key Concepts

Numerical Expressions

A numerical expression is essentially a combination of numbers and operational symbols, such as addition, subtraction, multiplication, and division. These expressions can also include exponents, which are a way of indicating repeated multiplication of a number by itself. The main goal of dealing with numerical expressions is to find their value by following the order of operations. This order ensures that the expression is simplified correctly by following certain steps or rules. When exponents are involved, they must be resolved before the basic operations of multiplication and division.

• Numerical expressions can be as simple as a single number or more complex, including multiple numbers with operations and exponents.
• Understanding these expressions is crucial because they form the basis for more complex algebraic expressions.

Exponentiation

Exponentiation refers to the process of raising a base number to the power of an exponent. In this scenario, the number is multiplied by itself a certain number of times, depending on the exponent value. For example, the expression \(a^b\) means that \(a\) is multiplied by itself \(b\) times. Exponents are a form of shorthand notation that makes it easier to work with very large or very small numbers.

• The base is the number being multiplied.
• The exponent indicates how many times the base is used as a factor.

The Zero Exponent Rule is a special rule in exponentiation. This rule asserts that any non-zero base raised to the power of zero equals one. For instance, \((-\frac{3}{4})^{0}\) equals 1 because of this rule. This simplifies calculations and helps in solving equations with ease.

Simplifying Expressions

Simplifying numerical expressions means reducing them to their simplest form by applying the order of operations, which usually refers to multiplying or dividing numbers within the expression before adding or subtracting them. Exponent rules also assist in simplifying expressions, particularly when terms have the same base. This can significantly decrease the complexity.

• Simplification aims to make expressions manageable and easy to interpret.
• It involves performing operations systematically based on pre-established rules.

In the provided example, simplifying \((-\frac{3}{4})^{0}\) involves applying the Zero Exponent Rule directly. This simplifies the expression instantly to 1, as any non-zero number's power of zero evaluates to one. Understanding the power of simplification helps make solving math problems more efficient and insightful.