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Multiply Rational Expression Calculator + Online Solver With Free Steps
A Multiply Rational Expressions Calculator is used to calculate the product of two simple or complex rational fractions. Solving rational fractions is a time-consuming and tiresome task. This online calculator makes this task easy and quick.
A Rational Expression can be written in the form of a fraction and is recurring or terminating in nature. This calculator can easily be used to apply Mathematical Functions by simply inserting the expressions into the fraction.
The calculator acts and the result is displayed on the output window. The result shows a detailed step-by-step solution leading to an answer in the form of a simple rational fraction.
What Is a Multiply Rational Expressions Calculator?
A Multiply Rational Expressions Calculator is an online calculator that can be used to solve the multiplication and division of rational expressions.
It can solve easy as well as hard mathematical and arithmetic operations by simply entering the fractions into the calculator.
This calculator works in your browser and uses the internet to perform the given mathematical problems efficiently. It multiplies and divides rational fractions in the same way as other numerical fractions are solved. However, it reduces the time required to solve such functions.
The Multiply Rational Expressions Calculator is designed to perform simple mathematical operations written in the form of correct rational expressions.
You can input both fractions into the calculator in the given boxes labeled Numerator and Denominator. The product and quotient of the entered rational fractions are displayed on the output screen as simple answers as well as detailed solutions.
How To Use the Multiply Rational Expressions Calculator?
To use a Multiply Rational Expressions Calculator, you should first set the rational fractions you want to solve. Enter the rational fractions into the calculator as instructed through the titles visible on the input screen. The calculator performs the operations and displays the result in another tab.
The following steps should be followed to use the online Multiply Rational Expressions Calculator:
Step 1
The calculator displays Enter first rational expression written above the input boxes of the first fraction and Enter second rational expression above the input boxes of the second fraction.
Step 2
Enter the numerator of the first fraction in the space given next to the title Enter the Numerator.
Step 3
Enter the denominator of the first fraction in the space given next to the title Enter the Denominator.
Step 4
Enter the numerator of the second fraction in the box in front of the title Enter the Numerator.
Step 5
Enter the denominator of the first fraction in the box titled Enter the Denominator.
Step 6
There is a box in the center with options of times divided by. Select the option based on the operation you want to perform.
Step 7
Press Calculate to view the answer.
Step 8
The output window displays the solution in two separate boxes. First, the input expression is written in product or quotient form. Second, the block titled Result shows the simplified rational expression.
Step 9
The result can also be viewed in detailed steps for easy understanding. The solution can also be observed in other forms.
Step 10
You can solve many such problems by entering the numbers into the calculator again and again.
It should be noted that the Multiply Rational Expressions Calculator can be used to calculate the product or quotient of rational expressions ranging from simple numerical fractions to complex rational expressions having variables in exponential form.
How Does a Multiply Rational Expressions Calculator Work?
A Multiply Rational Expressions Calculator works by taking the rational expressions in the form of fractions and multiplying or dividing them. It works similarly as doing it manually, except for all the lengthy calculations. The two rational expressions are divided or multiplied by taking the Least Common Factor (LCM) of the denominators. The calculator skips the hefty steps and displays the following things on the output screen:
Input Interpretation
The input interpretation interprets the problem entered into the calculator. The rational expressions are written in parenthesis in product or division form.
Results
This heading shows all the steps in detail that are required to operate on the fractions. The solution is also displayed in complete steps and more than one form as well.
What Is a Rational Expression?
A Rational Expression is a ratio between two polynomials. A polynomial is an expression in which the variable has an integer exponent, for instance $x^3+3x^2-1$. The polynomials are written in the form of a ratio between a and b i.e. a/b.
Simple mathematical operations like multiplication and division can be easily performed on rational expressions like other polynomials. The result of applying these operations to rational expressions produces a rational expression as a result as well.
The Domain of Rational Expressions
The domain of rational expressions can be any polynomial except the one that makes the denominator zero as it gives an undefined answer. A fraction cannot be rational if the denominator is zero. For example, for a rational expression 3x+1/x-4, x should not be equal to 4 as it makes the denominator zero.
Arithmetic Operations Performed on Rational Expressions
The Multiply Rational Expressions Calculator performs the following mathematical operations on the rational expressions:
Multiplication Operation
The two expressions are multiplied together by the factorization method. The expression obtained is simplified and written in descending order.
Division Operation
The two rational expressions are divided by inverting the second fraction and then multiplying both the fractions. The expression is then simplified and written in descending order.
Multiplication and division of rational expressions are easy to perform as compared to other functions and an online calculator makes them even easier.
Irrational Expression
An Irrational Expression Fraction is non-recurring and non-terminating. Rational expressions cannot be represented in the form of a ratio between two polynomials i.e. they cannot be written in a/b form. An irrational algebraic expression cannot be written in the form of the division of two polynomials.
Arithmetic Operations can also be performed on irrational expressions. However, the product or quotient of two irrational expressions may or may not be irrational. An irrational expression is obtained by multiplying or dividing a rational expression with an irrational expression.
Solved Examples
Here are some of the solved problems of rational fractions. These examples will make the process of multiplying and dividing rational expressions clearer.
Example 1
Multiply the following fractions:
Fraction 1:
\[ \dfrac{x^2+1}{x+1} \]
Fraction 2:
\[ \dfrac{x^2+3x+2}{3x^2+3} \]
Solution
The given rational expressions can be multiplied using the Multiply rational expressions calculator.
First, enter both fractions into the calculator. The output window displays the results as:
Input Interpretation
\[ \left( \dfrac{x^2+1}{x+1} \right)\left( \dfrac{x^2+3x+2}{3x^2+3} \right) \]
Results
\[= \dfrac{(x^3+x+1)(5x^2+9x+9)}{3x} \]
\[ =\left(x^2+ \dfrac{1}{x}+1 \right) \left( \dfrac{5x^2}{3}+3x+3 \right) \]
After simplification, the following expression is achieved:
\[ =\dfrac{5x^4}{3}+3x^3+ \dfrac{14x^2}{3}+ \dfrac{14x}{3}+ \dfrac{3}{x}+6 \]
Answer in more forms is:
\[= \dfrac{5x^5+9x^4+14x^3+14x^2+9}{3x} \]
\[= \dfrac{5}{3} \left( x^2+ \dfrac{1}{x}+1 \right)+ 3x \left( x^2+ \dfrac{1}{x}+1 \right)+ 3 \left( x^2+ \dfrac{1}{x}+1 \right) \]
Hence, by multiplying $\dfrac{x^2+1}{x+1}$ and $ \dfrac{x^2+3x+2}{3x^2+3} $the answer obtained is:
\[= \dfrac{5x^4}{3}+3x^3+ \dfrac{14x^2}{3}+ \dfrac{14x}{3}+ \dfrac{3}{x}+6 \]
\[ =\dfrac{5}{3} \left( x^2+ \dfrac{1}{x}+1 \right)+ 3x \left( x^2+ \dfrac{1}{x}+1 \right)+ 3 \left( x^2+ \dfrac{1}{x}+1 \right) \]
Example 2
Consider the following rational expressions:
\[ f(x)=\dfrac{x+3}{x-5} \]
\[ f(x)=\dfrac{x+7}{x^2-1} \]
Calculate the quotient of the fractions given above.
Solution
Enter both fractions into the calculator and select the option of “divided by” in the calculator. The output window shows the following results:
Input Interpretation
\[ =\dfrac{x+ \dfrac{3}{x}-5}{x+ \dfrac{7}{x^2}-1} \]
Results
\[ =\dfrac{(x^2-5x+3)x}{x^3-x^2+7} \]
\[ =\dfrac{x((x-5)x+3)}{(x-1)x^2+7} \]
The simplified expression is:
\[ =\dfrac{x^3-5x^2+3x}{x^3-x^2+7} \]
Another form of answer is:
\[ =\dfrac{x}{ \dfrac{7}{x^2}+x-1}- \dfrac{5}{ \dfrac{7}{x^2}+x-1}+ \dfrac{3}{ \dfrac{7}{x^2}+x-1}x \]
So, by dividing $ \dfrac{x+3}{x-5} $ by $ \dfrac{x+7}{x^2-1}$ you will get:
\[ =\dfrac{x^3-5x^2+3x}{x^3-x^2+7} \] or \[ =\dfrac{x^3-5x^2+3x}{x^3-x^2+7} \]
Example 3
For the following rational expressions:
Expression 1:
\[f(x) = \dfrac{x^4+x^3+2}{9} \]
Expression 2:
\[f(x) = \dfrac{x^2-5x+2}{x-3} \]
Calculate the product using the Multiply rational expressions calculator.
Solution
For the rational fractions \[ =\dfrac{x^4+x^3+2}{9} \] and \[ =\dfrac{x^2-5x+2}{x-3} \] the calculators displays the solution as follows:
Input Interpretation
\[= \left(x^4+x^3+ \dfrac{2}{9} \right)\left( x^2-5x+ \dfrac{2}{x}-3 \right) \]
Results
\[= \dfrac{(9x^4+9x^3+2)(x^3-5x^2-3x+2)}{9x} \]
\[ =x^6-4x^5-8x^4-x^3+ \dfrac{20x^2}{9}- \dfrac{10x}{9}+ \dfrac{4}{9x}+ \dfrac{2}{3} \]
The final expression comes out to be:
\[ =\dfrac{9x^7-36x^6-72x^5-9x^4+20x^3-10x^2-6x+4}{9x} \]
It can also be written in another form:
\[ =\dfrac{2}{9} \left(x^2-5x+ \dfrac{2}{x}-3 \right)+ \left(x^2-5x+ \dfrac{2}{x}-3 \right)x^4+\left(x^2-5x+ \dfrac{2}{x}-3 \right)x^3 \]
So, the product of $ \dfrac{x^4+x^3+2}{9} $ and $ \dfrac{x^2-5x+2}{x-3}$ is:
\[= \dfrac{9x^7-36x^6-72x^5-9x^4+20x^3-10x^2-6x+4}{9x} \] or \[ \dfrac{2}{9} \left(x^2-5x+ \dfrac{2}{x}-3 \right)+ \left(x^2-5x+ \dfrac{2}{x}-3 \right)x^4+\left(x^2-5x+ \dfrac{2}{x}-3 \right)x^3 \]