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Chapter 5: Problem 107

Simplify each expression. (a) \(3 \cdot 5^{-1}\) (b) \((3 \cdot 5)^{-1}\)

Short Answer

Expert verified
The simplified form of (a) is \(\frac{3}{5}\). The simplified form of (b) is \(\frac{1}{15}\).

Step by step solution

01

Simplify expression (a)

For the expression (a) \(3 \cdot 5^{-1}\), you need to recognize that \(5^{-1}\) is the reciprocal of 5, which is \(\frac{1}{5}\). Thus, the expression becomes: \[3 \cdot \frac{1}{5}\].
02

Perform the multiplication for (a)

Now multiply 3 by \(\frac{1}{5}\):\[3 \cdot \frac{1}{5} = \frac{3}{5}\].
03

Simplify expression (b)

For the expression (b) \((3 \cdot 5)^{-1}\), start by simplifying inside the parentheses.\[3 \cdot 5 = 15\].
04

Apply the negative exponent for (b)

Now apply the negative exponent to the simplified inside of the parentheses:\[(15)^{-1}\] is the reciprocal, so it becomes:\[\frac{1}{15}\].

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Key Concepts

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

Negative Exponents
Negative exponents might seem tricky at first, but they are quite simple once you understand the concept. A negative exponent means you are dealing with the reciprocal of the base raised to the corresponding positive exponent. For example, given the expression \(5^{-1}\), what this really means is \(\frac{1}{5}\).
  • The negative sign in the exponent signals that you flip the base to the bottom part of a fraction.
  • You then change the exponent to positive.
  • So, \(a^{-n}\) translates to \(\frac{1}{a^n}\).
Now, let's see this in action with our problem. For \(5^{-1}\), simply rewrite it as \(\frac{1}{5}\).This simplifies our overall expression and gives us a clearer understanding of the problem.
Reciprocals
Reciprocals are fundamental in algebra, particularly when dealing with fractions and negative exponents. A reciprocal of a number is essentially what you multiply that number by to get 1. For example, the reciprocal of 5 is \(\frac{1}{5}\), because: \[5 \cdot \frac{1}{5} = 1\].
  • Reciprocals are crucial in simplifying expressions involving negative exponents.
  • The reciprocal of any number \(a\) is \(\frac{1}{a}\).
  • You also apply this concept when dealing with more complex fractions and algebraic expressions.
In exercise (b), \((3 \cdot 5)^{-1}\), begin by solving inside the parentheses. We get \[15\]. Then take the reciprocal of 15: \[\frac{1}{15}\]. This is how we achieve the simplified result.
Multiplication
Multiplication is a basic yet essential operation in algebra. When multiplying numbers, you combine their values. Here are a few key points:
  • Multiplication of whole numbers is straightforward: multiply the numbers together.
  • When multiplying a whole number by a fraction, multiply the whole number by the numerator, keeping the denominator the same.
In our problem, the expression \(3 \cdot 5^{-1}\) requires multiplication after converting the negative exponent. We rewrite it as \(3 \cdot \frac{1}{5}\).Now multiply:\[3 \cdot \frac{1}{5} = \frac{3}{5}\]. So, the product of 3 and \(\frac{1}{5}\) is \(\frac{3}{5}\). This highlights how multiplication works seamlessly together with other algebraic principles like reciprocals.

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