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Chapter 4: Problem 28

Sort these expressions into two groups so that the expressions in each group are equal to one another. $$ m^{3} \quad\left(\frac{1}{m}\right)^{3} \quad m^{-3} \quad\left(\frac{1}{m}\right)^{-3} \quad \frac{1}{m^{3}} \quad m \div m^{4} $$

Short Answer

Expert verified
Group 1: \(m^{3}, \left(\frac{1}{m}\right)^{-3}\). Group 2: \(\left(\frac{1}{m}\right)^{3}, m^{-3}, \frac{1}{m^{3}}, m \div m^{4}\).

Step by step solution

01

Simplify Each Expression

First, simplify every expression provided:- Expression 1: \(m^{3}\)- Expression 2: \(\left(\frac{1}{m}\right)^{3} = \frac{1}{m^{3}}\)- Expression 3: \(m^{-3} = \frac{1}{m^{3}}\)- Expression 4: \(\left(\frac{1}{m}\right)^{-3} = m^{3}\)- Expression 5: \(\frac{1}{m^{3}}\)- Expression 6: \(m \div m^{4} = m^{1-4} = m^{-3} = \frac{1}{m^{3}}\)
02

Group Equal Expressions

Now, group the expressions that are equal to each other:- Group 1: \(m^{3}\) and \(\left(\frac{1}{m}\right)^{-3}\)- Group 2: \(\left(\frac{1}{m}\right)^{3}, m^{-3}, \frac{1}{m^{3}}, m \div m^{4}\)

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Key Concepts

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

Simplifying Expressions
Simplifying expressions is a crucial skill in algebra that helps us make complex expressions more manageable. The goal is to rewrite the expression in its simplest form without changing its value. Let's look at how we can simplify the given expressions from the exercise:
  • Expression: \( m^{3} \) is already in its simplest form.
  • Expression: \( \frac{1}{m^{3}} \) is also already simplified.
  • Expression: \( m^{-3} \) can be simplified to \( \frac{1}{m^{3}} \).
  • Expression: \( \frac{1}{m} \right) \right>^{-3} \) can be simplified to \( m^{3} \).
  • Expression: \( m \right> m^{4} \) can be simplified using the exponent rule of division, resulting in \( m^{-3} = \frac{1}{m^{3}} \).
By simplifying each expression, we were able to group equal expressions more efficiently.
Negative Exponents
Negative exponents can initially seem tricky, but they follow straightforward rules. A negative exponent tells us to take the reciprocal (or 'flip') of the base. This means:
\[ a^{-n} = \frac{1}{a^{n}} \]
Applying this rule to our exercise:
  • Expression: \( m^{-3} \) becomes \( \frac{1}{m^{3}} \).
  • Expression: \( \frac{1}{m} \right) \right>^{-3} \) becomes \( m^{3} \).
  • Expression: \( m \right> m^{4} \) simplifies using both division of exponents and the negative exponent rule, resulting in \( m^{-3} = \frac{1}{m^{3}} \).
Understanding negative exponents helps us greatly in algebra, particularly when simplifying expressions.
Division of Exponents
Division of exponents follows the rule: when you divide powers with the same base, you subtract the exponents. The formula looks like this:
\[ \frac{a^{m}}{a^{n}} = a^{m-n} \]
Let's use this rule on the expression \( m \right> m^{4} \) from the exercise:
  • Here, the base \( m \) is the same, so we subtract the exponents: \( m^{1-4} \).
  • This simplifies to \( m^{-3} \), which we know from the negative exponent rule equals \( \frac{1}{m^{3}} \).
Simplifying division using exponent rules helps streamline expressions and makes it easier to group and compare them.

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Most popular questions from this chapter

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