Multiply the expressions and simplify. 3 x + 6 y ⋅ x y + 2 y 3 \frac{3x+6}{y}\cdot\frac{xy+2y}{3}

( x + 2 ) 2 \left(x+2\right)^2 ( x + 2 ) 2

( x − 2 ) 2 \left(x-2\right)^2 ( x − 2 ) 2

Divide each expression and write the quotient in its simplest form. y − 1 y + 2 y − 1 3 ( y + 2 ) \frac{y-1}{\frac{y+2}{\frac{y-1}{3\left(y+2\right)}}}

3 y − 1 \frac{3}{y-1} y − 1 3

What is the "keep, change, flip" method for dividing rational expressions?

The "keep, change, flip" method is a strategy for dividing rational expressions. It means you keep the first rational expression the same, change the division sign to multiplication, and flip the second rational expression (take its reciprocal). For example, dividing a b \div c d becomes a b \times d c . Then multiply across numerators and denominators, factor completely, and simplify by canceling common factors. This method works the same for rational expressions with variables as it does for rational numbers.

8. Rational Expressions and Functions

Multiplying and Dividing Rational Expressions

8. Rational Expressions and Functions

Multiplying and Dividing Rational Expressions: Videos & Practice Problems

Video Lessons

Topic summary

Multiplying and dividing rational expressions involves multiplying numerators and denominators directly, similar to rational numbers. Simplification is achieved by factoring expressions into prime factors, including variables, and canceling common terms. For division, use the "keep, change, flip" method: keep the first expression, change division to multiplication, and flip the second expression. Factoring techniques, such as recognizing difference of squares, help simplify expressions. Understanding terms like numerator, denominator, quotient, and divisor is essential for working with multivariable polynomials and rational expressions effectively.

concept

Multiplying Rational Expressions

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Problem

Multiply the expressions and simplify.
$\frac{x}{3} \cdot \frac{2}{y}$

$x y over 6$

$x over 3 y$

$\frac{2}{3}$

$\frac{2x}{3y}$

Problem

Multiply the expressions and simplify.
$\frac{x}{x + 2} \cdot \frac{x + 2}{3}$

$\frac{1}{3}$

$\frac{x}{x + 2}$

$\frac{x}{3}$

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

Problem

Multiply the expressions and simplify.
$\frac{3 x + 6}{y} \cdot \frac{x y + 2 y}{3}$

$x minus 2$

$x plus 2$

$$\left$(x+2$\right$)^2$

$$\left$(x-2$\right$)^2$

example

Multiplying Rational Expressions Example 1

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concept

Dividing Rational Expressions

Video duration:

Problem

Divide each expression and write the quotient in its simplest form.
$\frac{8 x^{3}}{(2 x)^{5}} \div \frac{4 x^{4}}{16 x^{2}}$

$x^{4}$

$\frac{1}{x^{4}}$

$\frac{1}{4x^6}$

$4x^6$

Problem

Divide each expression and write the quotient in its simplest form.
$\frac{5 p + 5}{8 - 10 p} \div \frac{3 p + 3}{2 (8 - 10 p)}$

$\frac{10}{3}$

$\frac{3}{10}$

$$\frac$53$

$$\frac$13$

Problem

Divide each expression and write the quotient in its simplest form.
$\frac{y - 1}{\frac{y + 2}{\frac{y - 1}{3 (y + 2)}}}$

$\frac{1}{3}$

$\frac{3}{y + 2}$

$3$

$\frac{3}{y-1}$

example

Dividing Rational Expressions Example 2

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Here’s what students ask on this topic:

To multiply rational expressions with variables, multiply the numerators together and the denominators together, just like with rational numbers. For example, if you have $\frac{a}{b} $\times$ \frac{c}{d}$ , multiply across to get $\frac{a}{\cdot} \frac{b}{\cdot}$ . Then, factor both numerator and denominator completely, including factoring variables (e.g., $x^{2} = x \cdot x$ ). Cancel any common factors to simplify the expression. This process is similar to multiplying rational numbers but includes variable terms. Factoring helps identify common factors for cancellation, making the expression simpler.

The "keep, change, flip" method is a strategy for dividing rational expressions. It means you keep the first rational expression the same, change the division sign to multiplication, and flip the second rational expression (take its reciprocal). For example, dividing $\frac{a}{b} $\div$ \frac{c}{d}$ becomes $\frac{a}{b} $\times$ \frac{d}{c}$ . Then multiply across numerators and denominators, factor completely, and simplify by canceling common factors. This method works the same for rational expressions with variables as it does for rational numbers.

After multiplying or dividing rational expressions, simplify by factoring both the numerator and denominator completely. This includes factoring numbers into primes and factoring polynomials (e.g., difference of squares: $t^{2} - 1 = (t + 1)(t - 1)$ ). Then, cancel any common factors that appear in both numerator and denominator. Simplifying reduces the expression to its simplest form, making it easier to work with and understand. Recognizing common factors and factoring techniques is key to effective simplification.

Yes, multiplying and dividing rational expressions follows the same principles as with rational numbers. For multiplication, multiply numerators and denominators directly across. For division, use the "keep, change, flip" method: keep the first expression, change division to multiplication, and flip the second expression. The main difference is that rational expressions often include variables, so factoring and simplifying require handling polynomial expressions. Factoring helps identify common factors to cancel, simplifying the result. Understanding this similarity makes working with rational expressions more intuitive.

Factoring techniques that help simplify rational expressions include factoring out the greatest common factor (GCF), factoring trinomials, and recognizing special products like the difference of squares. For example, the difference of squares formula is $a^{2} - b^{2} = (a + b)(a - b)$ . Factoring polynomials into simpler binomials or monomials allows you to cancel common factors in the numerator and denominator. This process is essential for simplifying rational expressions after multiplication or division, ensuring the expression is in its simplest form.

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Multiplying and Dividing Rational Expressions: Videos & Practice Problems

Multiplying Rational Expressions

Multiply the expressions and simplify.
$\frac{x}{3} \cdot \frac{2}{y}$

Multiply the expressions and simplify.
$\frac{x}{x + 2} \cdot \frac{x + 2}{3}$

Multiply the expressions and simplify.
$\frac{3 x + 6}{y} \cdot \frac{x y + 2 y}{3}$

Multiplying Rational Expressions Example 1

Dividing Rational Expressions

Divide each expression and write the quotient in its simplest form.
$\frac{8 x^{3}}{(2 x)^{5}} \div \frac{4 x^{4}}{16 x^{2}}$

Divide each expression and write the quotient in its simplest form.
$\frac{5 p + 5}{8 - 10 p} \div \frac{3 p + 3}{2 (8 - 10 p)}$

Divide each expression and write the quotient in its simplest form.
$\frac{y - 1}{\frac{y + 2}{\frac{y - 1}{3 (y + 2)}}}$

Dividing Rational Expressions Example 2

Here’s what students ask on this topic:

How do you multiply rational expressions with variables?

What is the "keep, change, flip" method for dividing rational expressions?

How do you simplify rational expressions after multiplying or dividing?

Can you multiply and divide rational expressions the same way as rational numbers?

What factoring techniques help simplify rational expressions?

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Multiplying and Dividing Rational Expressions: Videos & Practice Problems

Multiplying Rational Expressions

Multiply the expressions and simplify.x3⋅2y\(\frac{x}{3}\[\cdot\]\frac{2}{y}\)

Multiply the expressions and simplify.xx+2⋅x+23\(\frac{x}{x+2}\[\cdot\]\frac{x+2}{3}\)

Multiply the expressions and simplify. 3x+6y⋅xy+2y3\(\frac{3x+6}{y}\[\cdot\]\frac{xy+2y}{3}\)

Multiplying Rational Expressions Example 1

Dividing Rational Expressions

Divide each expression and write the quotient in its simplest form.8x3(2x)5÷4x416x2\(\frac{8x^3}{(2x)^5}\[\div\]\frac{4x^4}{16x^2}\)

Divide each expression and write the quotient in its simplest form.5p+58−10p÷3p+32(8−10p)\(\frac{5p+5}{8-10p}\[\div\]\frac{3p+3}{2(8-10p)}\)

Divide each expression and write the quotient in its simplest form.y−1y+2y−13(y+2)\(\frac{y-1}{\frac{y+2}{\frac{y-1}{3\left(y+2\right)}\)}}

Dividing Rational Expressions Example 2

Here’s what students ask on this topic:

How do you multiply rational expressions with variables?

How do you multiply rational expressions with variables?

What is the "keep, change, flip" method for dividing rational expressions?

What is the "keep, change, flip" method for dividing rational expressions?

How do you simplify rational expressions after multiplying or dividing?

How do you simplify rational expressions after multiplying or dividing?

Can you multiply and divide rational expressions the same way as rational numbers?

Can you multiply and divide rational expressions the same way as rational numbers?

What factoring techniques help simplify rational expressions?

What factoring techniques help simplify rational expressions?

Your Intermediate Algebra tutors

Multiply the expressions and simplify.
$\frac{x}{3} \cdot \frac{2}{y}$

Multiply the expressions and simplify.
$\frac{x}{x + 2} \cdot \frac{x + 2}{3}$

Multiply the expressions and simplify.
$\frac{3 x + 6}{y} \cdot \frac{x y + 2 y}{3}$

Divide each expression and write the quotient in its simplest form.
$\frac{8 x^{3}}{(2 x)^{5}} \div \frac{4 x^{4}}{16 x^{2}}$

Divide each expression and write the quotient in its simplest form.
$\frac{5 p + 5}{8 - 10 p} \div \frac{3 p + 3}{2 (8 - 10 p)}$

Divide each expression and write the quotient in its simplest form.
$\frac{y - 1}{\frac{y + 2}{\frac{y - 1}{3 (y + 2)}}}$