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Chapter 4: Problem 15

Reduce each of the following rational expressions to lowest terms. $$\frac{x^{3}}{x}$$

Short Answer

Expert verified
\(x^2\)

Step by step solution

01

Identify the given rational expression

The rational expression given is \(\frac{x^3}{x}\).
02

Simplify the numerator and denominator

The numerator is \(x^3\) and the denominator is \(x\).
03

Apply the rules of exponents

Remember that \(\frac{a^m}{a^n} = a^{m-n}\) when \(a eq 0\). Here, \(\frac{x^3}{x} = x^{3-1} = x^2\).
04

Write the simplified form

The simplified form of the rational expression \(\frac{x^3}{x}\) is \(x^2\).

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Key Concepts

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

rules of exponents
Understanding the rules of exponents is crucial when simplifying rational expressions. An exponent tells you how many times to multiply a base number by itself. For example, in the expression \(\frac{a^m}{a^n}\), the base is 'a' and the exponents are 'm' and 'n'.
According to the rules of exponents:
\[ \frac{a^m}{a^n} = a^{m-n} \] Therefore, when you divide terms with the same base, you simply subtract the exponent in the denominator from the exponent in the numerator.
For instance, \(\frac{x^3}{x}\) can be simplified as \[ x^{3-1} = x^2 \] It's important to apply these rules correctly to avoid mistakes in simplification.
numerator and denominator simplification
Simplifying the numerator and denominator is an essential step. The numerator is the top part of the fraction, and the denominator is the bottom part.
In the exercise \(\frac{x^3}{x}\), the numerator is \(x^3\) and the denominator is \(x\). Before applying any rules, clearly identify each part. This helps to ensure you simplify correctly.
Remember, the goal in simplification is to reduce the fraction to its simplest form.
Here, using the rules of exponents from the previous section, we subtract the exponent in the denominator from the exponent in the numerator: \[ x^{3-1} = x^2 \] This produces a much simpler expression.
lowest terms
Putting a fraction into its lowest terms means making it as simple as possible. This is important for easier understanding and further calculations.
In our exercise example \(\frac{x^3}{x}\), we've simplified it down to \(x^2\). This is now in its lowest terms because there are no common factors in the numerator and denominator that can be further reduced.
Always ensure to double-check if an expression can be simplified further.
Simplifying to the lowest terms not only tidies up the expression but also makes it much easier to work with in future calculations and equations.

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Most popular questions from this chapter

In each of the following exercises, perform the indicated operations. Express your answer as a single fraction reduced to lowest terms. $$5 y^{3}-\frac{2}{y}+\frac{1}{2 y}$$

In each of the following exercises, perform the indicated operations. Express your answer as a single fraction reduced to lowest terms. $$\frac{11 c}{60 a^{2} b}-\frac{2 c}{45 a b^{3}}$$

In each of the following exercises, perform the indicated operations. Express your answer as a single fraction reduced to lowest terms. $$\frac{4 y}{3 x^{2}}-\frac{3}{2 x}+\frac{y}{x^{2}}$$

If the exercise is an equation, solve it; if not, perform the indicated operations and express your answer as a single fraction. $$\frac{x+5}{2}-\frac{x-1}{4}=2$$

Set up an equation or inequality and solve the problem. Be sure to indicate clearly what quantity your variable represents. Round to the nearest tenth where necessary. On a certain day the minimum temperature was \(14^{\circ} \mathrm{F},\) and the maximum temperature was \(23^{\circ} \mathrm{F}\). What was the temperature range in degrees Celsius? \(\left(\text { Use } F=\frac{9}{5} \mathrm{C}+32 .\right)\)
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