Posts Tagged ‘Rational’

Multiplying Rational Expressions

Multiplying Rational Expressions

 

Step 1:-  Factor all numerators and denominators as completely as possible.

 

Step 2:-  Apply the fundamental property.

 

Step 3 :- Multiply remaining factors in the numerator and remaining factors in the denominator. Leave the denominator in factored form.

 

Step 4:-  Check to be sure the product is in lowest terms.

Fundamental Property of Rational Numbers

Fundamental Property of Rational Numbers

 

If \farc{a}{b}is a rational number and if c is any nonzero real number, then

\frac{a}{b} = \frac{ac}{bc}.

 

In words, the numerator and denominator of a rational number may

either be multiplied or divided by the same nonzero number without

changing the value of the rational number.

Sets of Numbers

Sets of Numbers

 

Natural numbers or counting numbers

 

 {1, 2, 3, 4, 5, 6, . . . }

 

Whole numbers  {0, 1, 2, 3, 4, 5, 6, . . . }

 

Integers  {. . . , 23, 22, 21, 0, 1, 2, 3, . . . }

 

Rational numbers  \left \{ \frac{p}{q} \mid p  and  q   are    integers, q   \neq  0   \right \}

Examples: , 1.3, or \frac{4}{3}.

 

Irrational numbers  {x | x is a real number that is not rational}

Examples: \sqrt{3}

 

Real numbers {x | x is represented by a point on a

number line}*

 

Simplifying Complex Rational Expressions

Simplifying Complex Rational Expressions

A complex rational expression, or complex fraction expression, is a rational

expression that has one or more rational expressions within its numerator or

denominator. Here are some examples:

 

\frac{1+\frac{2}{x}}{3}

There are two methods to simplify complex rational expressions. We will

consider them both.

 

Method 1

 

Multiplying by the LCM of all the Denominators

To simplify a complex rational expression:

a. First, find the LCM of all the denominators of all the rational

expressions occurring within both the numerator and the

denominator of the complex rational expression.

b. Then multiply by 1 using LCM LCM.

c. If possible, simplify by removing a factor of 1.

 

Method 2

Adding in the Numerator and the Denominator

To simplify a complex rational expression:

a. Add or subtract, as necessary, to get a single rational expression in

the numerator.

b. Add or subtract, as necessary, to get a single rational expression in

the denominator.

c. Divide the numerator by the denominator.

d. If possible, simplify by removing a factor of 1.

 

Subtracting Rational Expressions

Subtracting Rational Expressions

We subtract rational expressions as we do rational numbers.

To subtract when the denominators are the same, subtract the numerators and keep the same denominator. Then simplify if possible.

Subtract  \frac{8}{x}-\frac{3}{x}

\frac{8}{x}-\frac{3}{x}=\frac{8-3}{x}=\frac{5}{x}

To subtract rational expressions with different denominators:

a. Find the LCM of the denominators. This is the least common

denominator (LCD).

b. For each rational expression, find an equivalent expression with

the LCD. To do so, multiply by 1 using a symbol for 1 made up of

factors of the LCD that are missing from the original denominator.

c. Subtract the numerators. Write the difference over the LCD.

d. Simplify if possible.

 

Adding Rational Expressions

Adding Rational Expressions

Adding Rational Expression with Like Denominators

 

To add when the denominators are the same, add the numerators and keep the same denominator. Then simplify if possible.

Example

\frac{x}{x+1}+\frac{2}{x+1}+\frac{x+2}{x+1}.

 

To add rational expressions with different denominators:

a. Find the LCM of the denominators. This is the least common

denominator (LCD).

b. For each rational expression, find an equivalent expression with

the LCD. To do so, multiply by 1 using an expression for 1 made up

of factors of the LCD that are missing from the original

denominator.

c. Add the numerators. Write the sum over the LCD.

d. Simplify if possible.

 

Rational Expressions and Replacements

Rational Expressions and Replacements

 

Rational numbers are quotients of integers. Some examples are

 

\frac{2}{3} , \frac{4}{-5} , \frac{-8}{17}.

The following are called rational expressions or fraction expressions. They are quotients, or ratios, of polynomials:

 

\frac{3}{4}, \frac{z}{6}, \frac{3}{2 + x}

 

A rational expression is also a division. For example,

 

\frac{3}{4} means 3 ÷ 4 and \frac{x-8}{x+2} means (x-8) ÷ (x+2).

 

Because rational expressions indicate division, we must be careful to avoid denominators of zero. When a variable is replaced with a number that produces a denominator equal to zero, the rational expression is not defined.

Rational Inequalities

Rational Inequalities

We adapt the preceding method when an inequality involves rational expressions.

We call these rational inequalities.

 

To solve a rational inequality:

a. Change the inequality symbol to an equals sign and solve the

related equation.

b. Find the numbers for which any rational expression in the

inequality is not defined.

c. Use the numbers found in steps (a) and (b) to divide the number

line into intervals.

d. Substitute a number from each interval into the inequality. If the

number is a solution, then the interval to which it belongs is part

of the solution set.

e. Select the intervals for which the inequality is satisfied and write

set-builder or interval notation for the solution set.

 

Negative Rational Exponents

Negative Rational Exponents

 

Negative rational exponents have a meaning similar to that of negative integer

exponents.

 a-m/n

For any rational number m/n and any positive real number a,

a-m/n means

\frac{1}{ a^{\frac{m}{n}}}

that is, am/n and a-m/n are reciprocals.

 

Rational Exponents

Rational Exponents

Expressions like  a1/2 , 5-1/4 ,and (2y)4/5 have not yet been defined. We will define

such expressions so that the general properties of exponents hold.

Consider a1/2.a1/2. If we want to multiply by adding exponents, it must

follow that a1/2 . a1/2 = a1/2 +1/2, or a1 Thus we should define a1/2 to be a

square root of a. Similarly, a1/3.a1/3 a1/3 = a1/3+1/3+1/3,or  a1,  so a1/3

should be defined to mean \sqrt[3]{a}.

 

a1/n

For any nonnegative real number a and any natural number index n

(n ≠1)

a1/n  means  \sqrt[n]{a} (the nonnegative nth root of a).

am/n

For any natural numbers m and n (n ≠1)and any nonnegative real

number a,

am/n means  \sqrt[n]{a^{m}} , or (\sqrt[n]{a})^{m}