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Chapter 9: Problem 1

Reduce each of the following fractions as completely as possible. $$\frac{3 x}{9 x^{2}}$$

Short Answer

Expert verified
\[ \frac{1}{3x} \]

Step by step solution

01

Identify the common factors

Examine the numerator and the denominator to identify any common factors. Here, the numerator is 3x and the denominator is 9x^2. Both share common factors of 3 and x.
02

Factor out the common terms

Write the fraction in terms of its factored form. The numerator 3x can be written as 3 * x, and the denominator 9x^2 can be written as 3 * 3 * x * x. So, the fraction becomes: \[ \frac{3 \times x}{3 \times 3 \times x \times x} \]
03

Cancel out the common factors

Now, cancel the common factors in the numerator and the denominator. \[ \frac{3 \times x}{3 \times 3 \times x \times x} = \frac{1}{3 \times x} \]
04

Simplify the fraction

What remains is \[ \frac{1}{3x} \].

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Key Concepts

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

Common Factors
When simplifying fractions, the first step is to find common factors between the numerator and the denominator.
Common factors are numbers or variables that both parts of the fraction share. For example, in the fraction \(\frac{3x}{9x^2}\), both the numerator (3x) and the denominator (9x虏) have common factors of 3 and x.
By identifying these common factors, we can simplify and eventually reduce the fraction.
Factoring
Once we've identified the common factors, the next step is factoring. Factoring means breaking down numbers or terms into their simplest parts, or 'factors.'
In our example, 3x can be factored into 3 * x, and 9x虏 can be factored into 3 * 3 * x * x.
This gives us:
\[ \frac{3 \times x}{3 \times 3 \times x \times x} \]
Factoring makes it easier to see the common terms and prepare for canceling.
Canceling Terms
After factoring, we can cancel out common terms in the numerator and the denominator.
Canceling terms means removing the same term from both the top and bottom of the fraction, since any number divided by itself equals 1.
In our example:
\[ \frac{3 \times x}{3 \times 3 \times x \times x} = \frac{1}{3 \times x} \]
When we cancel 3 and x from both the numerator and the denominator, we are left with \(\frac{1}{3x}\).
Fraction Reduction
The final step is fraction reduction. This means ensuring the fraction is in its simplest form.
Fraction reduction involves canceling all possible common factors until there are no more common factors left.
From our previous steps, we simplified \(\frac{3x}{9x^2}\) to \(\frac{1}{3x}\).
In this simplified form, the fraction can鈥檛 be reduced any further because there are no more common factors.
Thus, \(\frac{1}{3x}\) is the simplest and reduced form of the original fraction.

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Most popular questions from this chapter

An insect colony is growing according to the equation $$ n=\frac{800(2 t+3)}{0.2 t+5} $$ where \(n\) is the number of insects in the colony \(t\) hours after the initial formation of the colony. How many hours does it take until the colony has 2300 insects?

In Exercises \(1-36,\) solve each of the equations or inequalities explicitly for the indicated variable. $$m a+r=4 a-s \text { for } a$$

If the exercise is an equation, solve it and check. Otherwise, perform the indicated operations and simplify. $$\frac{1}{x+3} \div \frac{x+1}{x^{2}-9}$$

Solve each of the problems algebraically. Be sure to label what the variable represents. At a certain college the ratio of those who graduate in four years to those who take longer to graduate is 13 to \(3 .\) If in the last five years 2600 students graduated in 4 years, how many graduates were there altogether during the last five years?

If the exercise is an equation, solve it and check. Otherwise, perform the indicated operations and simplify. $$\frac{9}{x^{2}-3 x+2}-\frac{2}{x-1}=\frac{1}{x-2}$$
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