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Chapter 5: Problem 74

Simplify each expression. See Example \(10 .\) $$ -2 x^{0} $$

Short Answer

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Step by step solution

01

Understand the exponent

Any term that contains an exponent of zero, such as \(x^0\), equals 1 regardless of the base. This is because any non-zero number to the power of zero is always one. So, \(x^0 = 1\).
02

Substitute the simplified exponent

Replace \(x^0\) with 1 in the expression. Thus, \(-2x^0\) becomes \(-2 imes 1\).
03

Simplify the multiplication

Perform the multiplication with the replaced value. Multiply \(-2\) by 1, resulting in \(-2\).

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Key Concepts

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

Zero Exponent Rule
The zero exponent rule is an important concept in algebra, dealing with powers of zero. It's surprisingly simple: any non-zero number raised to the power of zero is equal to 1. This means that no matter how large or small a number is, if you see it as something like \[x^0 = 1\]or\[a^0 = 1\]it always simplifies to 1. Remember, this rule only applies to non-zero bases, since even raising zero to the power of zero is considered an indeterminate form in mathematics. Therefore, when you encounter expressions like \[-2x^0\]we substitute \(x^0\) with 1 according to the zero exponent rule.
Algebraic Expressions
Algebraic expressions are the backbone of algebra and provide a way to represent numbers and operations abstractly. They consist of variables, constants, and operators (such as addition and multiplication). In the expression\[-2x^0,\]"-2" is the constant part, and "x" is the variable part.
The role of expressions is to allow flexibility in mathematical modeling and problem-solving.
  • Variables: Represent numbers that can change or vary.
  • Constants: Fixed numbers included in the expression.
  • Operators: Indicate the operations to be performed (+, -, *, /).
While the expression may look complex at first, recognizing individual components like exponents can simplify things greatly, especially when using rules like the zero exponent rule.
Multiplication
The operation of multiplication is one of the basic arithmetic operations and involves adding a number to itself a specified number of times. In algebra, multiplication appears across expressions in different forms, including between constants and variables.

In the expression \[-2 \times 1\]multiplication simplifies the expression to\[-2.\]This shows that multiplying any number by 1 keeps the number unchanged, which is a simple yet crucial property in algebra.
  • When multiplying a negative number by a positive one, the result is negative, which explains our simplified result in this case.
  • It's important to be comfortable with multiplication because it's pervasive in algebraic operations.
Understanding multiplication within the context of expressions helps in simplifying and solving algebraic problems efficiently.

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Most popular questions from this chapter

Multiply. See Example 7. $$ (b+3)\left(2 b^{2}+b-3\right) $$

Evaluate each expression using exponential rules. Write each result in standard notation. See Example 7. $$ \frac{1.4 \times 10^{-2}}{7 \times 10^{-8}} $$

Multiply. $$(t+3)\left(t^{2}-5 t+5\right)$$

Multiply each expression. $$ 2 a\left(a^{2}+1\right) $$

Perform the indicated operations. $$(3 x-1)-(10 x-6)$$
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