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Chapter 6: Problem 14

Simplify each complex fraction. Use either method. $$ \frac{\frac{a}{x}}{\frac{a^{2}}{2 x}} $$

Short Answer

Expert verified
\frac{2}{a}

Step by step solution

01

Write the complex fraction as a single fraction

The given complex fraction can be written as: \[ \frac{\frac{a}{x}}{\frac{a^2}{2x}} \rightarrow \frac{a}{x} \times \frac{2x}{a^2} \] Rewriting a complex fraction as a multiplication problem.
02

Simplify the expression

Multiply the numerators together and the denominators together: \[ \frac{a \times 2x}{x \times a^2} = \frac{2ax}{a^2x} \] Cancel out common factors.
03

Cancel common factors

Notice that both the numerator and the denominator have a common factor of x and a: \[ \frac{2ax}{a^2x} = \frac{2a ot{x}}{a \times a ot{x}} = \frac{2}{a} \]

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Key Concepts

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

Complex Fractions
Complex fractions are fractions where the numerator, the denominator, or both contain a fraction themselves. They might seem complicated, but they can be simplified using various methods. One way to simplify a complex fraction is to rewrite it as a multiplication problem. For example, the complex fraction \[ \frac{\frac{a}{x}}{\frac{a^{2}}{2x}} \] can be rewritten to look simpler. This is a key step to make the problem more manageable.
Multiplication of Fractions
Once a complex fraction is rewritten as a multiplication problem, you multiply the numerators together and the denominators together. For instance, in the given problem: \[ \frac{\frac{a}{x}}{\frac{a^2}{2x}} \] can be rewritten as \[ \frac{a}{x} \times \frac{2x}{a^2} \]. This is because dividing by a fraction is the same as multiplying by its reciprocal. When you multiply fractions, just multiply across the numerators and across the denominators: \[ \frac{a \times 2x}{x \times a^2} = \frac{2ax}{a^2x} \].
Canceling Common Factors
Canceling common factors helps in simplifying the expression further. If the numerator and the denominator share common factors, you can divide them out. In this example: \[ \frac{2ax}{a^2x} \] both the numerator and the denominator have the common factors 'a' and 'x'. Canceling these out, we get: \[ \frac{2ax}{a^2x} = \frac{2a \cancel{x}}{a \times a \cancel{x}} = \frac{2}{a} \]. This reduces the fraction significantly.
Rational Expressions
Rational expressions are just fractions that contain polynomials in the numerator, the denominator, or both. Simplifying rational expressions follows the same principles as simplifying any other fractions. When working with rational expressions, look for common factors and make use of multiplication and division rules to simplify the expression. Practice and familiarity with these steps will help in handling even the most complex fractions with ease. For example, simplifying \[ \frac{2ax}{a^2x} \] to \[ \frac{2}{a} \] involves understanding that you're dealing with the polynomial factors 'a' and 'x'.

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Most popular questions from this chapter

Solve each formula or equation for the specified variable. $$ \frac{5}{p}+\frac{2}{q}+\frac{3}{r}=1 \text { for } r $$

Let \(P\), \(Q\), and \(R\) be rational expressions defined as follows. $$P=\frac{6}{x+3}, \quad Q=\frac{5}{x+1}, \quad R=\frac{4 x}{x^{2}+4 x+3}$$ Solve the equation \(P+Q=R\).

Solve each formula or equation for the specified variable. $$ I=\frac{E}{R+r} \text { for } R $$

The fractions here are continued fractions. Simplify by starting at "the bottom" and working upward. $$ r+\frac{r}{4-\frac{2}{6+2}} $$

Simplify each expression, using only positive exponents in the answer. $$ \frac{a^{-2}-4 b^{-2}}{3 b-6 a} $$
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