Algebraic Fractions Multiplication Division Practice Problems

Master multiplying and dividing algebraic fractions with step-by-step practice problems. Learn factorization, finding solution sets, and simplifying expressions.

📚Master Algebraic Fraction Operations with Interactive Practice
  • Factor algebraic expressions using common factors and short multiplication formulas
  • Find solution sets by identifying values that make denominators zero
  • Multiply algebraic fractions by simplifying numerators and denominators
  • Convert division problems to multiplication using reciprocal method
  • Simplify complex algebraic fraction expressions step-by-step
  • Apply factorization techniques to solve real algebraic fraction problems

Understanding Multiplication and Division of Algebraic Fractions

Complete explanation with examples

Multiplication and Division Operations in Algebraic Fractions

When we want to multiply or divide algebraic fractions, we will use the same tools that we use for the multiplication or division of common fractions with some small differences.

Steps to carry out for the multiplication of algebraic fractions 1 1 :

  • Let's try to extract the common factor.
    This can be the variable or any free number.
  • If this is not enough, we will factorize with short multiplication formulas or with trinomials.
  • Let's find the solution set.
    • How is the solution set found?
      We will make all the denominators we have equal to 0 0 and find the solution.
      The solution set will be X X : different from what causes our denominator to equal zero.
  • Let's simplify the fractions with determination.
  • Multiply numerator by numerator and denominator by denominator as in any fraction.
Detailed explanation

Practice Multiplication and Division of Algebraic Fractions

Test your knowledge with 19 quizzes

Select the domain of the following fraction:

\( \frac{8+x}{5} \)

Examples with solutions for Multiplication and Division of Algebraic Fractions

Step-by-step solutions included
Exercise #1

Complete the corresponding expression for the denominator

12ab?=1 \frac{12ab}{?}=1

Step-by-Step Solution

Let's examine the problem:

12ab?=1 \frac{12ab}{?}=1 Now let's think logically, and remember the known fact that dividing any number by itself always yields the result 1,

Therefore, in order to get the result 1 from dividing two numbers, the only way is to divide the number by itself, meaning-

The missing expression in the denominator of the fraction on the left side is the complete expression that appears in the numerator of the same fraction:

12ab 12ab .

Therefore- the correct answer is answer D.

Answer:

12ab 12ab

Video Solution
Exercise #2

Determine if the simplification below is correct:

4⋅84=18 \frac{4\cdot8}{4}=\frac{1}{8}

Step-by-Step Solution

We will divide the fraction exercise into two multiplication exercises:

44×81= \frac{4}{4}\times\frac{8}{1}=

We simplify:

1×81=8 1\times\frac{8}{1}=8

Therefore, the described simplification is false.

Answer:

Incorrect

Video Solution
Exercise #3

Determine if the simplification shown below is correct:

77⋅8=8 \frac{7}{7\cdot8}=8

Step-by-Step Solution

Let's consider the fraction and break it down into two multiplication exercises:

77×18 \frac{7}{7}\times\frac{1}{8}

We simplify:

1×18=18 1\times\frac{1}{8}=\frac{1}{8}

Therefore, the described simplification is false.

Answer:

Incorrect

Video Solution
Exercise #4

Determine if the simplification below is correct:

5⋅88⋅3=53 \frac{5\cdot8}{8\cdot3}=\frac{5}{3}

Step-by-Step Solution

Let's consider the fraction and break it down into two multiplication exercises:

88×53 \frac{8}{8}\times\frac{5}{3}

We simplify:

1×53=53 1\times\frac{5}{3}=\frac{5}{3}

Answer:

Correct

Video Solution
Exercise #5

Determine if the simplification below is correct:

6⋅36⋅3=1 \frac{6\cdot3}{6\cdot3}=1

Step-by-Step Solution

We simplify the expression on the left side of the approximate equality:

6̸⋅3̸6̸⋅3̸=?1↓1=!1 \frac{\textcolor{red}{\not{6}}\cdot\textcolor{blue}{\not{3}}}{\textcolor{red}{\not{6}}\cdot\textcolor{blue}{\not{3}}}\stackrel{?}{= }1\\ \downarrow\\ 1\stackrel{!}{= }1 therefore, the described simplification is correct.

Therefore, the correct answer is A.

Answer:

Correct

Video Solution

Frequently Asked Questions

Everything you need to know about Multiplication and Division of Algebraic Fractions

How do you multiply algebraic fractions step by step?

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First, factor both numerator and denominator using common factors or short multiplication formulas. Then find the solution set by setting denominators equal to zero. Finally, simplify by canceling common factors and multiply numerator by numerator, denominator by denominator.

What is the solution set in algebraic fractions?

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The solution set lists all values that the variable cannot equal. These are values that would make any denominator equal to zero, which would make the fraction undefined.

How do you divide algebraic fractions?

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Convert division to multiplication by keeping the first fraction unchanged, changing the division sign to multiplication, and flipping the second fraction (reciprocal). Then follow the same steps as multiplying algebraic fractions.

What factorization methods work for algebraic fractions?

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Use these methods in order: 1) Extract common factors from numerator and denominator, 2) Apply short multiplication formulas like difference of squares, 3) Factor trinomials when possible. Always factor completely before simplifying.

Why do we need to find the solution set before solving?

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The solution set identifies restricted values that make denominators zero. These values must be excluded from the final answer since division by zero is undefined in mathematics.

What are common mistakes when multiplying algebraic fractions?

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Common errors include: forgetting to factor completely, not finding the solution set, canceling terms instead of factors, and multiplying denominators incorrectly. Always factor first, then simplify by canceling common factors.

How do you simplify complex algebraic fraction expressions?

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Follow this process: 1) Factor all parts completely, 2) Identify and cancel common factors between numerator and denominator, 3) State the solution set, 4) Write the final simplified form. Never cancel terms that are added or subtracted.

When should you use short multiplication formulas in algebraic fractions?

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Use short multiplication formulas when you see patterns like a²-b² (difference of squares), a²+2ab+b² (perfect square trinomial), or a²-2ab+b² (perfect square trinomial). These formulas help factor expressions that common factor extraction cannot handle.

More Multiplication and Division of Algebraic Fractions Questions

Click on any question to see the complete solution with step-by-step explanations

Factorization and Algebraic Fractions

Solve Complex Rational Equation: ((2x-1)²)/(x-2) + ((x-2)²)/(2x-1) = 4.5x Calculate A and B in the Equation: Solving \((\sqrt{x}-\sqrt{x+1})/(x+1)=1\) Solve the Equation: 1/(x-2)² + 1/(x-2) = 1 Step-by-Step Solve the Rational Equation: (x³+1)/(x-1)² = x+4 Domain of 6/x: Determining Valid Input Values for a Fraction

Continue Your Math Journey

Master Multiplication and Division of Algebraic Fractions first, then advance to these related topics that build on your fraction skills

Suggested Topics to Practice in Advance

Factoring using contracted multiplicationFactorizationExtracting the common factor in parenthesesFactorization: Common factor extractionFactoring Trinomials

Topics Learned in Later Sections

Algebraic FractionsSimplifying Algebraic FractionsFactoring Algebraic FractionsAddition and Subtraction of Algebraic FractionsSolving Equations by FactoringUses of Factorization