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Chapter 2: Problem 4

Evaluate. Assume the variable does not equal zero. $$(5 m)^{0}$$

Short Answer

Expert verified
The expression \((5m)^{0}\) can be evaluated by applying the exponent rule, which states that any non-zero number raised to the power of 0 is equal to 1. Therefore, \((5m)^{0}=1\).

Step by step solution

01

Apply the exponent rule

To evaluate the expression (5m)^0, we will apply the rule that any non-zero number (in this case, 5m) raised to the power of 0 is equal to 1.
02

Find the value

Since (5m)^0 just involves applying the exponent rule, we can write: (5m)^0 = 1 So, the value of the given expression is 1.

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Key Concepts

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

Evaluate Expressions
Evaluating expressions is like solving a puzzle, where each piece has its unique role in reaching the final answer. When asked to evaluate an algebraic expression such as \((5m)^0\), the goal is to simplify the expression and find its numerical value. This process is essential in mathematics because it allows you to understand and manipulate expressions effectively.

To evaluate expressions, you need to follow certain steps:
  • First, identify and apply any mathematical rules or properties that may simplify the expression. This might include operations such as addition, subtraction, multiplication, or the use of exponents.
  • Second, perform any calculations necessary to arrive at the simplest form of the expression.
  • Finally, if the expression contains variables, substitute any given values to find the specific numerical result.
By practicing how to evaluate expressions, you gain confidence in handling various mathematical complexities, making problem-solving a smoother process.
Zero Exponent Rule
The zero exponent rule is a fascinating concept in mathematics that often surprises beginners. It states that any non-zero base raised to the power of zero equals one. In the expression \((5m)^0\), applying this rule gives us a result of 1, regardless of the value of \(m\) (as long as \(m\) is not zero).

Here's why the zero exponent rule makes sense. Consider how exponents work:
  • Exponents indicate repeated multiplication of the base. For example, \(5^3 = 5 \times 5 \times 5\).
  • Now, think of \(5^1\) as \(5\), and \(5^0\) should logically be divided by \(5^1\). Following this reasoning results in \(5^0 = \frac{5^1}{5^1} = 1\).
  • This logic applies universally to any non-zero base, hence the rule.
Understanding and utilizing the zero exponent rule is crucial in algebra, as it simplifies expressions and helps avoid potential errors when dealing with exponents.
Algebraic Expressions
An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation symbols. They are like codes that represent numbers and relationships between them. For example, \(5m\) is an algebraic expression where 5 is the coefficient, and \(m\) is the variable.

Algebraic expressions serve several purposes:
  • They allow us to generalize mathematical problems and relationships; instead of dealing with specific numbers, we can work with formulas that apply to many situations.
  • Expressions form the foundation for equations and functions, providing a basis for solving complex problems.
  • They are used to model real-world situations, making them critical in fields like engineering, science, and economics.
Being comfortable with algebraic expressions means understanding how to manipulate and simplify them, as seen in the zero exponent rule application to \((5m)^0\), which shows the potential for reduction to a simpler form, often a fundamental step in more complex algebraic procedures.

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Most popular questions from this chapter

Simplify using the quotient rule. $$\frac{15 w^{2}}{w^{10}}$$

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Write each number in scientific notation. -0.076

Perform the operation as indicated. Write the final answer without an exponent. \(\left(-3 \times 10^{-2}\right)\left(-2.6 \times 10^{-3}\right)\)

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