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Chapter 5: Problem 48

Simplify each expression. $$ 75 b^{0} $$

Short Answer

Expert verified
The simplified expression is 75.

Step by step solution

01

Understanding Exponents

The zero exponent rule states that any number raised to the power of zero is equal to 1. This means that for any term \(b^{0}\), the result is 1 as long as \(b\) is not zero.
02

Applying the Zero Exponent Rule

Apply the zero exponent rule to simplify the expression \(75b^{0}\). Since \(b^{0} = 1\), the expression becomes \(75 \times 1\).
03

Simplifying the Expression

Multiply 75 by 1. This gives us the final simplified expression, which is 75.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Expressions
In algebra, simplifying expressions is about transforming complex statements into simpler forms, while maintaining their original value. The main goal is to make it easier to interpret or solve. When you see an expression like \(75b^{0}\), using rules such as the zero exponent rule is crucial. It helps in reducing the expression to its simplest form. Let's break this down further:
  • Identify components: In the expression \(75b^{0}\), we have a number multiplied by a variable raised to the zero power.
  • Apply the appropriate rule: Here, the zero exponent rule tells us how to handle \(b^{0}\).
  • Perform arithmetic operations: Multiplying by 1 simplifies the expression to just 75.
Recognizing and applying such rules speed up algebraic problem-solving. It transforms a seemingly complex task into something straightforward, saving time and reducing errors.
Exponents
Exponents are mathematical notations indicating how many times a number, known as the base, is multiplied by itself. If you see \(b^n\), it means you multiply \(b\) by itself \(n\) times. The zero exponent rule is a special scenario, simplifying these understandings:
  • Zero Exponent Rule: Any non-zero number raised to the power of zero equals 1, so \(b^{0} = 1\).
  • Positive Exponents: Indicate expansion, such as \(b^2 = b \times b\).
  • Negative Exponents: Represent reciprocal values, like \(b^{-1} = \frac{1}{b}\).
Understanding these rules makes it easier to handle and simplify expressions involving exponents, ensuring clarity and precision in handling algebraic tasks.
Algebra Basics
Algebra is a foundational branch of mathematics focused on variables and number operations. It's about finding the unknowns and forming relationships between them using expressions and equations. Let's dive into some core principles:
  • Variables: Symbols, often letters, that represent unknown numbers. In our example, \(b\) is the variable.
  • Expressions: Combinations of variables, numbers, and operators (like \(+\), \(-\), \(\times\)) that represent a specific value.
  • Simplification: The process of transforming an expression to its simplest form without changing its value, making it easier to work with.
By mastering these basics, concepts like the zero exponent rule become second nature. Strengthening your grasp on algebra makes more advanced topics accessible, paving the way for success in more complex mathematical fields.

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