Multiply & Divide Rational Expressions

Topics

Try VividMath Premium to unlock full access

Start Free Trial

Time limit: 0

Quiz summary

0 of 6 questions completed

Questions:

Information

–

You have already completed the quiz before. Hence you can not start it again.

Quiz is loading...

You must sign in or sign up to start the quiz.

You have to finish following quiz, to start this quiz:

Lets see your results!

Points

Score

Time

Retry Quiz

Answered
Review

Question 1 of 6

1. Question

Multiply

\frac{3 x^{2}}{4 x} \times \frac{5 x^{2}}{10 x^{3}}

$(3x^2)/(4x) xx (5x^2)/(10x^3)$

1. $\frac{3 x^{8}}{8}$ $(3x^8)/8$
2. $\frac{5}{8}$ $5/8$
3. $40$ $40$
4. $\frac{3}{8}$ $3/8$

Hint

Help Video

Correct

Fantastic!

Incorrect

Need Text

Play

Current Time 0:00

Duration Time 0:00

Remaining Time -0:00

Stream TypeLIVE

Loaded: 0%

Progress: 0%

0:00

Fullscreen

Video Acceleration:

Off

00:00

Mute

Playback Rate

2x
1.5x
1.25x
1x
0.75x
0.5x

Subtitles

subtitles off

Captions

captions off
English

Chapters

Chapters

Multiplying Rational Expressions

\frac{a}{b} \times \frac{c}{d} = \frac{a b}{c d}

$a/b xx c/d = (ab)/(cd)$

Multiply the terms in the numerator and then in the denominator.

$\frac{3 x^{2}}{4 x} \times \frac{5 x^{2}}{10 x^{3}}$ $(3x^2)/(4x) xx (5x^2)/(10x^3)$	$=$ $=$	$\frac{3 x^{2} \times 5 x^{2}}{4 x \times 10 x^{3}}$ $(3x^2 xx 5x^2)/(4x xx 10x^3)$

	$=$ $=$	$\frac{(3 \times 5) x^{2 + 2}}{(4 \times 10) x^{1 + 3}}$ $((3xx5)x^(2+2))/((4xx10) x^(1+3)$	Multiply the constants and the variables

	$=$ $=$	$\frac{15 x^{4}}{40 x^{4}}$ $(15x^4)/(40x^4)$	Apply Multiplication Rule of Exponents

	$=$ $=$	$\frac{15}{40}$ $15/40$
	$=$ $=$	$\frac{3}{8}$ $3/8$	Express in lowest terms

\frac{3}{8}

$3/8$

Question 2 of 6

2. Question

Multiply

\frac{m^{2} - 5 m - 6}{m - 2} \times \frac{4 m - 8}{m - 6}

$(m^2-5m-6)/(m-2) xx (4m-8)/(m-6)$

1. $4 (m + 1)$ $4(m+1)$
2. $4$ $4$
3. $m + 1$ $m+1$
4. $4 (m - 1)$ $4(m-1)$

Hint

Help Video

Correct

Excellent!

Incorrect

Need Text

Play

Current Time 0:00

Duration Time 0:00

Remaining Time -0:00

Stream TypeLIVE

Loaded: 0%

Progress: 0%

0:00

Fullscreen

Video Acceleration:

Off

00:00

Mute

Playback Rate

2x
1.5x
1.25x
1x
0.75x
0.5x

Subtitles

subtitles off

Captions

captions off
English

Chapters

Chapters

Multiplying Rational Expressions

\frac{a}{b} \times \frac{c}{d} = \frac{a b}{c d}

$a/b xx c/d = (ab)/(cd)$

Look for polynomials that can be factorised before proceeding with the operation. Since the numerator of the first term is in standard form

(

$($ $a$ $a$

x^{2} +

$x^2+$ $b$ $b$

x +

$x+$ $c$ $c$

= 0)

$=0)$ we can factorise using the cross method.

m^{2}

$m^2$ $- 5$ $-5$

m

$m$ $- 6$ $-6$

= 0

$=0$

To factorise, we need to find two numbers that add to $- 5$ $-5$ and multiply to $- 6$ $-6$

- 6

$-6$ and

1

$1$ fit both conditions

$- 6 + 1$ $-6+1$	$=$ $=$	$- 5$ $-5$
$- 6 \times 1$ $-6 xx 1$	$=$ $=$	$- 6$ $-6$

Read across to get the factors.

(m - 6) (m + 1) = 0

$(m-6)(m+1)=0$

Do the same for the numerator of the second term.

4 m - 8

$4m-8$

=

$=$

4 (m - 2)

$4(m-2)$

Rewrite the expression with the factors.

$\frac{m^{2} - 5 m - 6}{m - 2} \times \frac{4 m - 8}{m - 6}$ $(m^2-5m-6)/(m-2) xx (4m-8)/(m-6)$	$=$ $=$	$\frac{(m - 6) (m + 1)}{m - 2} \times \frac{4 (m - 2)}{m - 6}$ $((m-6)(m+1))/(m-2) xx (4(m-2))/(m-6)$

	$=$ $=$	$(m + 1) \times 4$ $(m+1) xx 4$	Cancel out $m - 6$ $m-6$ and $m - 2$ $m-2$
	$=$ $=$	$4 (m + 1)$ $4(m+1)$	Simplify

4 (m + 1)

$4(m+1)$

Question 3 of 6

3. Question

Multiply

\frac{4 x^{2} + 4 x}{5 x^{2} - 5 x - 60} \times \frac{x^{2} - 6 x + 8}{x^{2} - 2 x}

$(4x^2+4x)/(5x^2-5x-60) xx (x^2-6x+8)/(x^2-2x)$

1. $\frac{4 (x + 2)}{5 (x + 3)}$ $(4(x+2))/(5(x+3))$
2. $\frac{4 (x + 3)}{5 (x - 2)}$ $(4(x+3))/(5(x-2))$
3. $\frac{4}{5}$ $4/5$
4. $\frac{4 (x + 1)}{5 (x + 3)}$ $(4(x+1))/(5(x+3))$

Hint

Help Video

Correct

Nice Job!

Incorrect

Need Text

Play

Current Time 0:00

Duration Time 0:00

Remaining Time -0:00

Stream TypeLIVE

Loaded: 0%

Progress: 0%

0:00

Fullscreen

Video Acceleration:

Off

00:00

Mute

Playback Rate

2x
1.5x
1.25x
1x
0.75x
0.5x

Subtitles

subtitles off

Captions

captions off
English

Chapters

Chapters

Multiplying Rational Expressions

\frac{a}{b} \times \frac{c}{d} = \frac{a b}{c d}

$a/b xx c/d = (ab)/(cd)$

Look for polynomials that can be factorised before proceeding with the operation.

$4 x^{2} + 4 x$ $4x^2+4x$	$=$ $=$	$4 x (x + 1)$ $4x(x+1)$
$5 x^{2} - 5 x - 60$ $5x^2-5x-60$	$=$ $=$	$5 (x^{2} - x - 12)$ $5(x^2-x-12)$
$x^{2} - 2 x$ $x^2-2x$	$=$ $=$	$x (x - 2)$ $x(x-2)$

The denominator of the first term is in standard form

(

$($ $a$ $a$

x^{2} +

$x^2+$ $b$ $b$

x +

$x+$ $c$ $c$

= 0)

$=0)$ so we can factorise using the cross method.

x^{2}

$x^2$ $-$ $-$

x

$x$ $- 12$ $-12$

= 0

$=0$

To factorise, we need to find two numbers that add to $- 1$ $-1$ and multiply to $- 12$ $-12$

- 4

$-4$ and

3

$3$ fit both conditions

$- 4 + 3$ $-4+3$	$=$ $=$	$- 1$ $-1$
$- 4 \times 3$ $-4 xx 3$	$=$ $=$	$- 12$ $-12$

Read across to get the factors.

(x - 4) (x + 3) = 0

$(x-4)(x+3)=0$

The numerator of the second term is also in standard form

(

$($ $a$ $a$

x^{2} +

$x^2+$ $b$ $b$

x +

$x+$ $c$ $c$

= 0)

$=0)$ so we can factorise using the cross method.

x^{2}

$x^2$ $- 6$ $-6$

x +

$x+$ $8$ $8$

= 0

$=0$

To factorise, we need to find two numbers that add to $- 6$ $-6$ and multiply to $8$ $8$

- 4

$-4$ and

- 2

$-2$ fit both conditions

$- 4 - 2$ $-4-2$	$=$ $=$	$- 6$ $-6$
$- 4 \times - 2$ $-4 xx -2$	$=$ $=$	$8$ $8$

Read across to get the factors.

(x - 4) (x - 2) = 0

$(x-4)(x-2)=0$

Rewrite the expression with the factors.

$\frac{4 x^{2} + 4 x}{5 x^{2} - 5 x - 60} \times \frac{x^{2} - 6 x + 8}{x^{2} - 2 x}$ $(4x^2+4x)/(5x^2-5x-60) xx (x^2-6x+8)/(x^2-2x)$	$=$ $=$	$\frac{4 x (x + 1)}{5 (x - 4) (x + 3)} \times \frac{(x - 4) (x - 2)}{x (x - 2)}$ $(4x(x+1))/(5(x-4)(x+3)) xx ((x-4)(x-2))/(x(x-2))$

	$=$ $=$	$\frac{4 (x + 1)}{5 (x + 3)}$ $(4(x+1))/(5(x+3))$	Cancel out $x$ $x$ , $x - 4$ $x-4$ and $x - 2$ $x-2$

\frac{4 (x + 1)}{5 (x + 3)}

$(4(x+1))/(5(x+3))$

Question 4 of 6

4. Question

Divide

\frac{6 a^{2} + 11 a - 10}{3 a - 2} \div (2 a + 5)

$(6a^2+11a-10)/(3a-2) -: (2a+5)$

1. $3 a - 2$ $3a-2$
2. $1$ $1$
3. $2 a + 5$ $2a+5$
4. $2$ $2$

Hint

Help Video

Correct

Well Done!

Incorrect

Need Text

Play

Current Time 0:00

Duration Time 0:00

Remaining Time -0:00

Stream TypeLIVE

Loaded: 0%

Progress: 0%

0:00

Fullscreen

Video Acceleration:

Off

00:00

Mute

Playback Rate

2x
1.5x
1.25x
1x
0.75x
0.5x

Subtitles

subtitles off

Captions

captions off
English

Chapters

Chapters

Dividing Rational Expressions

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}

$a/b -: c/d = a/b xx d/c$

Since the numerator of the first term is in standard form

(

$($ $a$ $a$

x^{2} +

$x^2+$ $b$ $b$

x +

$x+$ $c$ $c$

= 0)

$=0)$ we can factorise using the cross method.

$6$ $6$

a^{2} +

$a^2+$ $11$ $11$

a

$a$ $- 10$ $-10$

= 0

$=0$

To factorise, we need to find two numbers that add to $11$ $11$ and multiply to $- 10$ $-10$

3 a

$3a$ ,

2 a

$2a$ ,

- 2

$-2$ and

5

$5$ fit both conditions

$15 a - 4 a$ $15a – 4a$	$=$ $=$	$11$ $11$ $a$ $a$
$- 2 \times 5$ $-2 xx 5$	$=$ $=$	$- 10$ $-10$

Read across to get the factors.

(3 a - 2) (2 a + 5) = 0

$(3a-2)(2a+5)=0$

Rewrite the expressions

$\frac{6 a^{2} + 11 a - 10}{3 a - 2} \div (2 a + 5)$ $(6a^2+11a-10)/(3a-2) -: (2a+5)$	$=$ $=$	$\frac{(3 a - 2) (2 a + 5)}{3 a - 2} \div (2 a + 5)$ $((3a-2)(2a+5))/(3a-2) -: (2a+5)$

	$=$ $=$	$\frac{(3 a - 2) (2 a + 5)}{3 a - 2} \times \frac{1}{2 a + 5}$ $((3a-2)(2a+5))/(3a-2) xx 1/(2a+5)$	Apply Division Formula

	$=$	$1$	$3a-2$ and $2a+5$ cancel out in the denominator

$1$

Question 5 of 6

5. Question

Divide

$(3x^2-9x)/(x^2-12x+36) -: (x^3-9x)/(x-6)$

1. $3/((x-6)(x+3))$
2. $(x-6)(x+3)$
3. $3$
4. $3/(x-6)$

Hint

Help Video

Correct

Great Work!

Incorrect

Need Text

Play

Current Time 0:00

Duration Time 0:00

Remaining Time -0:00

Stream TypeLIVE

Loaded: 0%

Progress: 0%

0:00

Fullscreen

Video Acceleration:

Off

00:00

Mute

Playback Rate

2x
1.5x
1.25x
1x
0.75x
0.5x

Subtitles

subtitles off

Captions

captions off
English

Chapters

Chapters

Dividing Rational Expressions

$a/b -: c/d = a/b xx d/c$

Since the denominator of the first term is in standard form

$($ $a$

$x^2+$ $b$

$x+$ $c$

$=0)$ we can factorise using the cross method.

$x^2$ $-12$

$x+$ $36$

$=0$

To factorise, we need to find two numbers that add to $-12$ and multiply to $36$

$-6$ and

$-6$ fit both conditions

$-6 – 6$	$=$	$-12$
$-6 xx -6$	$=$	$36$

Read across to get the factors.

$(x-6)(x-6)=0$

Factorise remaining expressions.

$3x^2-9x$	$=$	$3x(x-3)$	Factor $3x$ out
$x^3-9x$	$=$	$x(x^2-9)$	Factor $x$ out
	$=$	$x(x-3)(x+3)$	$x^2-9 = (x-3)(x+3)$

Rewrite the expressions

$(3x^2-9x)/(x^2-12x+36) -: (x^3-9x)/(x-6)$	$=$	$(3x^2-9x)/(x^2-12x+36) xx (x-6)/(x^3-9x)$	Apply Division Formula

	$=$	$(3x(x-3))/((x-6)(x-6)) xx (x-6)/(x(x-3)(x+3))$	Show factors

	$=$	$3/((x-6)(x+3))$	Cancel out like terms

$3/((x-6)(x+3))$

Question 6 of 6

6. Question

Divide

$(m^2+2m-8)/(m^2-3m+2) -: (m^2+5m+4)/(m^2-4m+3)$

1. $m+3$
2. $(m+3)/(m+1)$
3. $(m+1)/(m-3)$
4. $(m-3)/(m+1)$

Hint

Help Video

Correct

Correct!

Incorrect

Need Text

Play

Current Time 0:00

Duration Time 0:00

Remaining Time -0:00

Stream TypeLIVE

Loaded: 0%

Progress: 0%

0:00

Fullscreen

Video Acceleration:

Off

00:00

Mute

Playback Rate

2x
1.5x
1.25x
1x
0.75x
0.5x

Subtitles

subtitles off

Captions

captions off
English

Chapters

Chapters

Dividing Rational Expressions

$a/b -: c/d = a/b xx d/c$

Since the given polynomials are in standard form

$($ $a$

$x^2+$ $b$

$x+$ $c$

$=0)$ we can factorise using the cross method.

$m^2+$ $2$

$m$ $-8$

$=0$

To factorise, we need to find two numbers that add to $2$ and multiply to $-8$

$4$ and

$-2$ fit both conditions

$4 – 2$	$=$	$2$
$4 xx -2$	$=$	$-8$

Read across to get the factors.

$(m+4)(m-2)=0$

Do the same for the remaining polynomials.

$m^2$ $-3$

$m+$ $2$

$=0$

To factorise, we need to find two numbers that add to $-3$ and multiply to $2$

$-2$ and

$-1$ fit both conditions

$-2 – 1$	$=$	$-3$
$-2 xx -1$	$=$	$2$

Read across to get the factors.

$(m-2)(m-1)=0$

$m^2+$ $5$

$m+$ $4$

$=0$

To factorise, we need to find two numbers that add to $5$ and multiply to $4$

$4$ and

$1$ fit both conditions

$4 + 1$	$=$	$5$
$4 xx 1$	$=$	$4$

Read across to get the factors.

$(m+4)(m+1)=0$

$m^2$ $-4$

$m+$ $3$

$=0$

To factorise, we need to find two numbers that add to $-4$ and multiply to $3$

$-3$ and

$-1$ fit both conditions

$-3 – 1$	$=$	$-4$
$-3 xx -1$	$=$	$3$

Read across to get the factors.

$(m-3)(m-1)=0$

Rewrite the expressions

$(m^2+2m-8)/(m^2-3m+2) -: (m^2+5m+4)/(m^2-4m+3)$	$=$	$(m^2+2m-8)/(m^2-3m+2) xx (m^2-4m+3)/(m^2+5m+4)$	Apply Division Formula

	$=$	$((m+4)(m-2))/((m-2)(m-1)) xx ((m-3)(m-1))/((m+4)(m+1))$	Show factors

	$=$	$(m-3)/(m+1)$	Cancel out like terms

$(m-3)/(m+1)$

Quizzes

Lowest Common Denominator
Add & Subtract Rational Expressions
Add & Subtract Rational Expressions with Unlike Denominators 1
Add & Subtract Rational Expressions with Unlike Denominators 2
Multiply & Divide Rational Expressions