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Chapter 1: Problem 13

Multiply and simplify. Leave the answer in exponential notation. $$ m^{0} \cdot m^{8} $$

Short Answer

Expert verified
m^{8}

Step by step solution

01

Understand the Zero Exponent Rule

The zero exponent rule states that any non-zero number raised to the power of zero is 1. Thus, applying this rule, we get: \( m^{0} = 1 \)
02

Multiply the Expressions

Now, multiply 1 by \( m^{8} \). Since multiplying any number by 1 leaves the number unchanged, we get: 1 \cdot m^{8} = m^{8}
03

Write the Final Answer

Thus, the final simplified expression in exponential notation is: \( m^{8} \)

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Key Concepts

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

exponential notation
Exponential notation is a way to express repeated multiplication of the same number. For example, instead of writing \( m \times m \times m \times m \), we can write \( m^4 \). This notation makes it easier to understand and work with large numbers or repeated factors. Exponential notation consists of a base (the number being multiplied) and an exponent (the number of times the base is multiplied by itself).

In our exercise, the base is \( m \) and the exponents are 0 and 8 (\( m^0 \) and \( m^8 \)). Understanding exponential notation is essential for simplifying expressions in algebra.
multiplication of exponents
Multiplying exponents can be straightforward once you understand the rules that govern them. When you multiply expressions with the same base, you simply add the exponents. For example, \( m^a \times m^b = m^{a+b} \).

However, in our original exercise, we start with \( m^0 \times m^8 \). According to the zero exponent rule, any non-zero number raised to the power of zero is 1. Thus, \( m^0 = 1 \). After this step, we multiply \( 1 \times m^8 \).

Since multiplying any expression by 1 leaves it unchanged, the simplified expression remains \( m^8 \). Understanding these rules helps in quickly and accurately simplifying exponential expressions.
algebraic simplification
Algebraic simplification involves breaking down complex expressions into their simplest form. This process often includes applying rules such as the zero exponent rule, combining like terms, and factoring.

In our exercise, we used the zero exponent rule to simplify \( m^0 \) to 1. Then we multiplied 1 by \( m^8 \). Since multiplying by 1 does not change the value, we end up with \( m^8 \).

To simplify algebraic expressions correctly:
  • Identify and apply algebraic rules.
  • Combine like terms.
  • Simplify each step methodically.
This systematic approach ensures that you can handle more complex algebraic problems with ease.

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Most popular questions from this chapter

Write a sentence in which the word "factor" appears once as a verb and once as a noun.

Calculate using the rules for order of operations. If an expression is undefined, state this. $$ 19-\left(4+2 \cdot 3^{2}\right) $$

Calculate using the rules for order of operations. If an expression is undefined, state this. $$ (-4)^{2} $$

To the student and the instructor: Throughout this text selected exercises are marked with the icon Ahal. Students who pause to inspect an Ahal exercise should find the answer more readily than those who proceed mechanically. This may involve looking at an earlier exercise or example, or performing calculations in a more efficient manner. Some Ahal exercises are left unmarked to encourage students to always pause before working a problem. Evaluate each expression using the values provided. $$ [3(a-b)]^{2}, \text { for } a=7 \text { and } b=5 $$

Translate to an algebraic expression. \(\\{x | x=5 n, n \text { is a natural number }\\}\)
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