Algebraic Rational Expressions

In summary, the restrictions on the variables are that $a$ and $b$ cannot both be 0, and that 2a^4b^2 cannot be 0.
  • #1
jjlittle00
2
0
I am attempting to find the solution to the following question.

Simplify and state the restrictions on the variables\(\displaystyle \frac{5a^5b^6}{10a^2b^3}\div\frac{2a^4b^2}{20a^3b^5}\)

Not really understanding how to find the restrictions with these set variables.
 
Last edited:
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  • #2
Hello and welcome to MHB, jjlittle00! (Wave)

I am assuming the expression is as follows:

\(\displaystyle \frac{5a^5b^6}{10a^2b^3}\div\frac{2a^2b^3}{20a^2b^3}\)

Before we proceed, is this correct?
 
  • #3
MarkFL said:
Hello and welcome to MHB, jjlittle00! (Wave)

I am assuming the expression is as follows:

\(\displaystyle \frac{5a^5b^6}{10a^2b^3}\div\frac{2a^4b^2}{20a^3b^5}\)

Before we proceed, is this correct?
Yes this is correct. Just made one small correction.
 
  • #4
jjlittle00 said:
Yes this is correct. Just made one small correction.

Okay, we now have:

\(\displaystyle \frac{5a^5b^6}{10a^2b^3}\div\frac{2a^4b^2}{20a^3b^5}\)

We have one rational expression being divided by another. In order for these expressions to be defined, we cannot have either denominator being equal to zero. What values of $a$ and/or $b$ will cause either denominator to be zero?
 
  • #5
Also, when we divide by a fraction, we "invert and multiply": [tex]\frac{5a^5b^6}{10a^2b^3}\frac{20a^3b^5}{2a^4b^2}[/tex]. So that, in addition to the requirement that the denominators of the original fractions not being 0, [tex]2a^4b^2[/tex] cannot be 0. That is effectively saying that a and b cannot be 0.

Of course, to "simplify" you cancel as many "a"s and "b"s, in numerator and denominator, as you can.
 
  • #6
HallsofIvy said:
Also, when we divide by a fraction, we "invert and multiply": [tex]\frac{5a^5b^6}{10a^2b^3}\frac{20a^3b^5}{2a^4b^2}[/tex]. So that, in addition to the requirement that the denominators of the original fractions not being 0, [tex]2a^4b^2[/tex] cannot be 0. That is effectively saying that a and b cannot be 0.

Of course, to "simplify" you cancel as many "a"s and "b"s, in numerator and denominator, as you can.

I was going to get to all that eventually...honest I was...:p
 

FAQ: Algebraic Rational Expressions

What is an algebraic rational expression?

An algebraic rational expression is a mathematical expression that contains variables, numbers, and operations such as addition, subtraction, multiplication, and division. The variables in the expression can have any value, and the expression can be simplified by canceling out common factors.

How is an algebraic rational expression different from a regular algebraic expression?

An algebraic rational expression includes fractions or rational numbers, whereas a regular algebraic expression does not. This means that in an algebraic rational expression, the variables can have any real number value, including fractions, decimals, and negative numbers, while in a regular algebraic expression, the variables are typically restricted to whole numbers.

What are the common operations used in simplifying algebraic rational expressions?

The most common operations used in simplifying algebraic rational expressions are factoring, canceling out common factors, and finding a common denominator for fractions. Factoring involves breaking down a polynomial expression into smaller, simpler expressions, while canceling out common factors simplifies the expression by reducing the number of terms. Finding a common denominator allows for easier addition and subtraction of fractions in the expression.

What are the key rules to remember when simplifying algebraic rational expressions?

There are a few key rules to remember when simplifying algebraic rational expressions. First, always look for common factors that can be canceled out. Second, when adding or subtracting fractions, find a common denominator before performing the operation. Third, when multiplying fractions, multiply the numerators and denominators separately. Finally, when dividing fractions, invert the second fraction and multiply it by the first fraction.

How can algebraic rational expressions be used in real-life situations?

Algebraic rational expressions can be used in various real-life situations, such as calculating proportions, solving for unknown quantities in equations, and determining rates of change. They are also commonly used in financial applications, such as calculating interest rates or determining monthly payments on loans. In science and engineering, algebraic rational expressions can be used to model and solve problems involving quantities that can change over time.

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