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Chapter 6: Problem 15

Find each product or quotient. Simplify your answers. $$ \frac{1}{a} \div \frac{1}{a} $$

Short Answer

Expert verified
The quotient is 1.

Step by step solution

01

Understand the Problem

Given the expression \(\frac{1}{a} \div \frac{1}{a}\), we need to find the quotient and simplify the answer.
02

Apply Division of Fractions

To divide fractions, multiply the first fraction by the reciprocal of the second fraction. That means we convert \(\frac{1}{a} \div \frac{1}{a}\) to \(\frac{1}{a} \times \frac{a}{1}\).
03

Multiply the Fractions

Now multiply \(\frac{1}{a}\) by \(\frac{a}{1}\): \(\frac{1}{a} \times \frac{a}{1} = \frac{1 \times a}{a \times 1} = \frac{a}{a} = 1\).
04

Simplify

Since \(\frac{a}{a} = 1\) for any non-zero value of \a\, the simplified answer is 1.

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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 fundamental skill in algebra and arithmetic. The goal is to express a mathematical expression in its simplest form. When you simplify an expression, you make it easier to understand and solve.

If we look at the original problem \(\frac{1}{a} \div \frac{1}{a}\), the goal is to simplify the fraction after division. The final simplified form is always easier to interpret.

  • Identify similar terms
  • Reduce fractions by dividing the numerator and the denominator by common factors
  • Combine like terms and constants as much as possible

In the given exercise, we simplify by using the properties of fractions and the concept of reciprocals.
reciprocal
Understanding reciprocals is essential in various mathematical operations, especially when dealing with fractions. The reciprocal of a number is simply one divided by that number. For a fraction, you swap the numerator and the denominator.

For example, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\).

In our exercise, the original division problem \(\frac{1}{a} \div \frac{1}{a}\) requires us to change the division into multiplication by using the reciprocal of \(\frac{1}{a}\):

  • The reciprocal of \(\frac{1}{a}\) is \(\frac{a}{1}\).
  • So, \(\frac{1}{a} \div \frac{1}{a}\) turns into \(\frac{1}{a} \times \frac{a}{1}\).
multiplication of fractions
Multiplication of fractions is straightforward and involves multiplying the numerators together and the denominators together. This process is essential for solving many fraction-related problems, including our exercise.

Here's a step-by-step explanation:

  • Take two fractions: \(\frac{a}{b}\) and \(\frac{c}{d}\).
  • Multiply the numerators: \a \times c\.
  • Multiply the denominators: \b \times d\.
  • Combine them to form the new fraction: \(\frac{a \times c}{b \times d}\).

In our exercise, we multiplied the two fractions \(\frac{1}{a} \times \frac{a}{1}\):

  • Multiplying the numerators: \1 \times a = a\.
  • Multiplying the denominators: \a \times 1 = a\.
  • The result is \(\frac{a}{a} = 1\).

This demonstrates how multiplication of fractions allows us to easily simplify the expression back to 1.

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Most popular questions from this chapter

Rewrite each expression using a single base and a single exponent. $$ a^{12} \cdot\left(a^{2}\right)^{-7} $$

Simplify each fraction. $$ \frac{15}{75} $$

Use the distributive property to rewrite each expression without parentheses. $$ -(3 n-4) $$

Expand each expression. Simplify your results by combining like terms. \(2(x+y)+x(3+y)(x+2)\)

Solve each equation. \(3.2-2 b=1.1\)
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