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Monday 20th of May
Calculations with Negative Numbers
Solving Linear Equations
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Algebra Expressions
Evaluating Expressions and Solving Equations
Fraction rules
Factoring Quadratic Trinomials
Multiplying and Dividing Fractions
Dividing Decimals by Whole Numbers
Adding and Subtracting Radicals
Subtracting Fractions
Factoring Polynomials by Grouping
Slopes of Perpendicular Lines
Linear Equations
Roots - Radicals 1
Graph of a Line
Sum of the Roots of a Quadratic
Writing Linear Equations Using Slope and Point
Factoring Trinomials with Leading Coefficient 1
Writing Linear Equations Using Slope and Point
Simplifying Expressions with Negative Exponents
Solving Equations 3
Solving Quadratic Equations
Parent and Family Graphs
Collecting Like Terms
nth Roots
Power of a Quotient Property of Exponents
Adding and Subtracting Fractions
Solving Linear Systems of Equations by Elimination
The Quadratic Formula
Fractions and Mixed Numbers
Solving Rational Equations
Multiplying Special Binomials
Rounding Numbers
Factoring by Grouping
Polar Form of a Complex Number
Solving Quadratic Equations
Simplifying Complex Fractions
Common Logs
Operations on Signed Numbers
Multiplying Fractions in General
Dividing Polynomials
Higher Degrees and Variable Exponents
Solving Quadratic Inequalities with a Sign Graph
Writing a Rational Expression in Lowest Terms
Solving Quadratic Inequalities with a Sign Graph
Solving Linear Equations
The Square of a Binomial
Properties of Negative Exponents
Inverse Functions
Rotating an Ellipse
Multiplying Numbers
Linear Equations
Solving Equations with One Log Term
Combining Operations
The Ellipse
Straight Lines
Graphing Inequalities in Two Variables
Solving Trigonometric Equations
Adding and Subtracting Fractions
Simple Trinomials as Products of Binomials
Ratios and Proportions
Solving Equations
Multiplying and Dividing Fractions 2
Rational Numbers
Difference of Two Squares
Factoring Polynomials by Grouping
Solving Equations That Contain Rational Expressions
Solving Quadratic Equations
Dividing and Subtracting Rational Expressions
Square Roots and Real Numbers
Order of Operations
Solving Nonlinear Equations by Substitution
The Distance and Midpoint Formulas
Linear Equations
Graphing Using x- and y- Intercepts
Properties of Exponents
Solving Quadratic Equations
Solving One-Step Equations Using Algebra
Relatively Prime Numbers
Solving a Quadratic Inequality with Two Solutions
Operations on Radicals
Factoring a Difference of Two Squares
Straight Lines
Solving Quadratic Equations by Factoring
Graphing Logarithmic Functions
Simplifying Expressions Involving Variables
Adding Integers
Factoring Completely General Quadratic Trinomials
Using Patterns to Multiply Two Binomials
Adding and Subtracting Rational Expressions With Unlike Denominators
Rational Exponents
Horizontal and Vertical Lines
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Multiplying and Dividing Fractions

Examples with Solutions

Example 1: Multiply

solution: This is just another way of posing the problem: Simplify:

Proceeding as in the previous example, we get

as the final solution. We could have done this last step as

to get the same final result, once factors are repositioned between the numerator and denominator to get rid of negative exponents.

Example 2: Simplify .

solution: This is very similar to the expressions handled in the first two examples. Proceeding in the same fashion, we get

(This is what the actual multiplication of the two fractions amounts to. Now this result must be simplified.)

(Here we expand the numerical factors into products of prime factors, and we also sort out the various literal factors.)

(Cancel the common numerical factors and combine powers of the literal symbols.)

(This is a fully simplified form, but it contains negative exponents.)

as the final result with negative exponents eliminated.

Example 3: Simplify .

solution: This is one fraction divided by another. Following the pattern given at the beginning of this document, we know that the first step here is to rewrite the expression as the first fraction multiplied by the reciprocal of the second fraction:

Now the remainder of the work is to simplify this multiplication, exactly as we dealt with the first three examples. So

as the final answer.

A very common error here is to start by correctly rewriting the original division problem as a multiplication, but then doing the multiply step in a totally bizarre way – numerators with denominators, as in

But this is totally wrong! When we multiply two fractions, it is always numerator times numerator and denominator times denominator, regardless of where the multiplication problem originally came from. If you examine the eventual result that would be obtained here, you’ll see that it amounts to what we would get if we had multiplied the two original fractions together, rather than dividing the first fraction by the second one. In other words, what has effectively been done by using this erroneous method is to change the original division symbol to a multiplication symbol. This must be an error. So, always remember: dividing by a fraction is equivalent to multiplying by its reciprocal, and multiplying is always done the same way, regardless of from where the original fractions were obtained.

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