Complex Fractions with Rational Expressions

​You will need to know how to..

• ​Simplify Rational Expressions

• ​Multiply Rational Expressions

• ​Divide Complex Rational Expressions

• ​Add and Subtract Rational Expressions

​Simplifying Rational Expressions

• ​Factor Completely

• ​Cancel out any matches in the numerator and denominator

• ​Remember when dividing with exponents you subtract them.

• ​You can only simplify if its being multiplied, unless you can divide the same thing out of ALL terms.

• ​The denominator cannot = 0 so whatever values would make the denominator 0 is excluded from the domain

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Multiple Choice

Reduce, or simplify

the rational expression.

• 3 x y³

• 8 x³ y

• 3 x³ y

• 8 x y³

Multiple Choice

Reduce, or simplify

the algebraic fraction.

• x - 3

• -21

• -3

• ( x - 3 ) / ( x )

Multiple Choice

Reduce, or simplify

the rational expression.

• ( x + 5 )( x - 5 ) / ( x + 5 )( x - 2 )

• ( x - 5 ) / ( x - 2 )

• ( x + 5 ) / ( x - 2 )

• -25 / ( 3x - 10 )

​Multiplying Rational Expressions

• ​Multiply straight across.

• ​Factor everything and cross off common factors in the numerator and denominator

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Multiple Choice

Multiply and Simplify

• 4(x+3)/(x+2),

• (x+2)/4(x+1)

• (x+2)/4(x+3),

• 4(x+1)/(x+2),

Multiple Choice

  $$\frac{3b^2-6b}{b^2+2b-3}\cdot\frac{b^2-8b+7}{b-2}$$  

• $$\frac{1}{b\left(b-2\right)};\ b\ne0,\ 2$$  

• $$\frac{b-3}{3b^2\left(b+6\right)};\ b\ne-6,\ 0$$  

• $$b-8;\ b\ =\ all\ real\ numbers$$  

• $$\frac{3b\left(b-7\right)}{b+3};\ b\ \ne-3,\ 1,\ 2$$  

Multiple Choice

$$\frac{24+5x-x^2}{x^2-5x-24}\cdot\frac{8x+24}{7x^2+21x}$$  

• $$7;\ x\ =\ all\ real\ numbers$$  

• $$\frac{x+5}{3x};\ x\ne0$$  

• $$21x;\ x\ =\ all\ real\ numbers$$  

• $$-\frac{8}{7x};\ x\ne-3,\ 0,\ 8$$  

​Dividing Complex Rational Expressions

• ​If it is complex fraction (fraction on top of another fraction) rewrite it our horizontally

• ​KEEP CHANGE FLIP!!

• ​Then follow rules for multiplication

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Multiple Choice

Divide and Simplify

• 1/6

• x+10/ 6x+60

• x+8/ 6x+60 

• (x+10)(x+8)/ 6x+60

Multiple Choice

Divide the Rational Expression

• A

• B

• C

• D

Multiple Choice

Simplify: $$\frac{\frac{5k^3-25k^2}{5k^2}}{\frac{k^2+2k-35}{9}}$$  

• $$\frac{4k^2}{k-7}$$  

• $$\frac{9}{k+7}$$  

• $$\frac{9}{4}$$  

• 1

​Adding and Subtracting Rational Expression

• ​You must have a common denominator.

• ​Whatever you multiply the bottom by, you must also multiply to the top

• ​Remember when subtracting you must change ALL the signs in the numerator of the second fraction

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Multiple Choice

Simplify the expression.

Multiple Choice

Add

• A.

• B.

• C.

• D.

Multiple Choice

Subtract: $$\frac{5x-3}{x+2}-\frac{3x-1}{x+2}$$  

• $$\frac{2x-4}{x+2}$$  

• $$\frac{8x-4}{x+2}$$  

• $$\frac{2x-2}{x+2}$$  

• $$\frac{2x-2}{0}$$  

Multiple Choice

Add: $$\frac{5}{x-2}+\frac{3}{x+7}$$  

• $$\frac{8}{\left(x-2\right)\left(x+7\right)}$$  

• $$\frac{8}{2x+5}$$  

• $$\frac{8x}{\left(x-2\right)\left(x+7\right)}$$  

• $$\frac{8x+29}{\left(x-2\right)\left(x+7\right)}$$  

Multiple Choice

$$Simplify:\ \frac{5a^3c^{10}}{20a^5c^4}$$  

• $$\frac{4c^6}{a^2}$$  

• $$\frac{1a^2c^6}{4}$$  

• $$\frac{1c^6}{4a^2}$$  

• $$\frac{1a^2}{4c^6}$$  

Multiple Choice

$$Simplify:\ \frac{-30x^5y^8}{6x^5y^9}$$  

• $$\frac{1}{5y}$$  

• $$-5y$$  

• $$-5x^{10}y^{17}$$  

• $$\frac{-5}{y}$$  

Multiple Choice

$$Simplify:\ \ \frac{x^2+6x}{x^2+3x-18}$$  

• $$\frac{x}{x+3}$$  

• $$\frac{x}{x-3}$$  

• $$\frac{x+6}{x-3}$$  

• $$\frac{x}{x+6}$$  

Multiple Choice

$$Simplify:\ \ \frac{x^2+7x+10}{x^2-4}$$  

• $$\frac{x-5}{x+2}$$  

• $$\frac{x-5}{x-2}$$  

• $$\frac{x+5}{x+2}$$  

• $$\frac{x+5}{x-2}$$  

Multiple Choice

$$Simplify:\ \ \frac{x^2-25}{x^2-6x+5}$$  

• $$\frac{x+5}{x-1}$$  

• $$\frac{x-5}{x-1}$$  

• $$\frac{x+5}{x+1}$$  

• $$\frac{x-5}{x+1}$$  

Multiple Choice

Add

• A.
• B.
• C.
• D.

Multiple Choice

Add

• A.
• B.
• C.
• D.

Multiple Choice

• 10 / 14a⁴

• 21 / 49a⁴

• 3 / a⁴

• 3 / 7a⁴

Multiple Choice

Divide

• A
• B
• C
• D

Multiple Choice

Multiply

• A
• B
• C
• D

Multiple Choice

Divide the following expressions.

• $$\frac{4\left(x-4\right)}{\left(x+5\right)}$$

• $$\frac{\left(x+4\right)^2\left(x-4\right)}{16\left(x+5\right)}$$

• Can't be simplified

• $$\frac{\left(x+5\right)}{4\left(x-4\right)\ }$$

Multiple Choice

Simplify completely.

$$\frac{1}{a-4}\div\frac{a-6}{5a^2-20a}$$  

• $$\frac{2a\left(a+7\right)}{a+3}$$  

• $$\frac{5a}{a-6}$$  

• $$a-7$$  

• $$\frac{a+7}{a+3}$$  

Multiple Choice

  $$\frac{3b^2-6b}{b^2+2b-3}\cdot\frac{b^2-8b+7}{b-2}$$  

• $$\frac{1}{b\left(b-2\right)};\ b\ne0,\ 2$$  

• $$\frac{b-3}{3b^2\left(b+6\right)};\ b\ne-6,\ 0$$  

• $$b-8;\ b\ =\ all\ real\ numbers$$  

• $$\frac{3b\left(b-7\right)}{b+3};\ b\ \ne-3,\ 1,\ 2$$  

Multiple Choice

$$\frac{24+5x-x^2}{x^2-5x-24}\cdot\frac{8x+24}{7x^2+21x}$$  

• $$7;\ x\ =\ all\ real\ numbers$$  

• $$\frac{x+5}{3x};\ x\ne0$$  

• $$21x;\ x\ =\ all\ real\ numbers$$  

• $$-\frac{8}{7x};\ x\ne-3,\ 0,\ 8$$  

Multiple Choice

$$\frac{n^2+4n-32}{n^2-9n+20}\cdot\frac{n^2-11n+30}{5n-30}$$  

• $$\frac{n+6}{16n^4};\ n\ne0$$  

• $$\frac{6n}{\left(n-4\right)\left(n+6\right)};\ n\ne-6,\ 4$$  

• $$\frac{3\left(n-8\right)}{n+8};\ n\ne-8$$  

• $$\frac{n+8}{5};\ n\ne4,\ 5,\ 6$$  

Multiple Choice

Simplify the expression.

Multiple Choice

Add the following rational expression.

• (7x+3)  /  (x-5)(x+8)
• (2x+3)  /  (x-5)(x+8)
• (2x+4)  /  (x-5)(x+8)
• (7x+4)  /  (x-5)(x+8)

Multiple Choice

Multiple Choice

Multiple Choice

Subtract and simplify. $$\frac{6}{a+2}-\frac{12}{a^2+6a+8}$$  

• $$\frac{6\left(a+2\right)}{a+4}$$  

• $$\frac{6}{a+4}$$  

• $$\frac{6a+12}{\left(a+2\right)\left(a+4\right)}$$  

• $$\frac{6\left(a+2\right)}{\left(a+2\right)\left(a+4\right)}$$  

Multiple Choice

• A

• B

• C

• D