Problem 11 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

1

Step 1: Identify the Power of Zero

The expression to simplify is \( \left(-\frac{3}{4}\right)^{0} \). It involves raising a number to the power of zero. According to the exponent rules, any non-zero number raised to the power of zero is 1.

Step 2: Simplify the Expression

Apply the rule: Any number raised to the power of zero is 1. Therefore, \( \left(-\frac{3}{4}\right)^{0} = 1 \).

Key Concepts

Numerical Expressions

A numerical expression is a combination of numbers, operations, and sometimes parentheses that represent a single value. For example, the expression \(-\frac{3}{4}\) is a fraction, and when combined with an exponent, becomes a full numerical expression. These expressions are foundational in algebra because they allow us to apply various mathematical operations to solve problems or simplify expressions.

Working with numerical expressions involves:

• Identifying the numbers involved.
• Understanding the operations applied (such as addition, subtraction, multiplication, division, and exponentiation).
• Applying the correct order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

By properly interpreting numerical expressions, students can simplify complex equations and they become more confident in solving mathematical problems efficiently.

Exponent Rules

Exponent rules are essential guidelines used when dealing with powers of numbers or variables. These rules help simplify expressions and solve equations more easily. Understanding these rules is crucial for anyone learning algebra.

Some fundamental exponent rules include:

• Product of Powers: When multiplying similar bases, keep the base the same and add the exponents. For instance, \(a^m \times a^n = a^{m+n}\).
• Quotient of Powers: When dividing similar bases, keep the base the same and subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
• Power of a Power: To raise a power to another power, multiply the exponents: \((a^m)^n = a^{m\cdot n}\).
• Power of a Product: The power applies to each factor in the product: \((ab)^n = a^n \cdot b^n\).

By applying these rules, it becomes easier to simplify and manipulate expressions involving exponents. Once mastered, handling even complex algebraic expressions becomes manageable.

Power of Zero

The power of zero rule is a simple yet essential concept in algebra. It states that any non-zero number raised to the zero power equals one. This seems surprising at first, but it becomes clearer upon understanding the logic behind it.

Why is this the case? Consider the quotient of powers rule: \(a^m / a^n = a^{m-n}\). If the exponents are equal (i.e., \(m=n\)), the result is \(a^{0} = 1\), as any number divided by itself is 1. This reinforces why any non-zero base raised to the power of zero equals 1.

This principle is widely applicable, not just limited to integers or fractions. Whether it's positive, negative, or a fraction as in the example \(\left(-\frac{3}{4}\right)^{0}\), they all simplify to 1 if the base is non-zero. Embracing this rule allows students to confidently navigate expressions where zero exponents appear, simplifying and solving them correctly.