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Multiply & Divide Rational ExpressionsMultiply & Divide Rational Expressions
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Question 1 of 6
1. Question
Multiply`(3x^2)/(4x) xx (5x^2)/(10x^3)`Hint
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Multiplying Rational Expressions
`a/b xx c/d = (ab)/(cd)`Multiply the terms in the numerator and then in the denominator.`(3x^2)/(4x) xx (5x^2)/(10x^3)` `=` `(3x^2 xx 5x^2)/(4x xx 10x^3)` `=` `((3xx5)x^(2+2))/((4xx10) x^(1+3)` Multiply the constants and the variables `=` `(15x^4)/(40x^4)` Apply Multiplication Rule of Exponents `=` `15/40` `=` `3/8` Express in lowest terms `3/8` -
Question 2 of 6
2. Question
Multiply`(m^2-5m-6)/(m-2) xx (4m-8)/(m-6)`Hint
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Multiplying Rational Expressions
`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``x^2+``b``x+``c``=0)` we can factorise using the cross method.`m^2``-5``m``-6``=0`To factorise, we need to find two numbers that add to `-5` and multiply to `-6``-6` and `1` fit both conditions`-6+1` `=` `-5` `-6 xx 1` `=` `-6` 
Read across to get the factors.`(m-6)(m+1)=0`Do the same for the numerator of the second term.`4m-8` `=` `4(m-2)` Rewrite the expression with the factors.`(m^2-5m-6)/(m-2) xx (4m-8)/(m-6)` `=` `((m-6)(m+1))/(m-2) xx (4(m-2))/(m-6)` `=` `(m+1) xx 4` Cancel out `m-6` and `m-2` `=` `4(m+1)` Simplify `4(m+1)` -
Question 3 of 6
3. Question
Multiply`(4x^2+4x)/(5x^2-5x-60) xx (x^2-6x+8)/(x^2-2x)`Hint
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Multiplying Rational Expressions
`a/b xx c/d = (ab)/(cd)`Look for polynomials that can be factorised before proceeding with the operation.`4x^2+4x` `=` `4x(x+1)` `5x^2-5x-60` `=` `5(x^2-x-12)` `x^2-2x` `=` `x(x-2)` The denominator of the first term is in standard form `(``a``x^2+``b``x+``c``=0)` so we can factorise using the cross method.`x^2``-``x``-12``=0`To factorise, we need to find two numbers that add to `-1` and multiply to `-12``-4` and `3` fit both conditions`-4+3` `=` `-1` `-4 xx 3` `=` `-12` 
Read across to get the factors.`(x-4)(x+3)=0`The numerator of the second term is also in standard form `(``a``x^2+``b``x+``c``=0)` so we can factorise using the cross method.`x^2``-6``x+``8``=0`To factorise, we need to find two numbers that add to `-6` and multiply to `8``-4` and `-2` fit both conditions`-4-2` `=` `-6` `-4 xx -2` `=` `8` 
Read across to get the factors.`(x-4)(x-2)=0`Rewrite the expression with the factors.`(4x^2+4x)/(5x^2-5x-60) xx (x^2-6x+8)/(x^2-2x)` `=` `(4x(x+1))/(5(x-4)(x+3)) xx ((x-4)(x-2))/(x(x-2))` `=` `(4(x+1))/(5(x+3))` Cancel out `x`, `x-4` and `x-2` `(4(x+1))/(5(x+3))` -
Question 4 of 6
4. Question
Divide`(6a^2+11a-10)/(3a-2) -: (2a+5)`Hint
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Dividing Rational Expressions
`a/b -: c/d = a/b xx d/c`Since the numerator of the first term is in standard form `(``a``x^2+``b``x+``c``=0)` we can factorise using the cross method.`6``a^2+``11``a``-10``=0`To factorise, we need to find two numbers that add to `11` and multiply to `-10``3a`, `2a`, `-2` and `5` fit both conditions`15a – 4a` `=` `11``a` `-2 xx 5` `=` `-10` 
Read across to get the factors.`(3a-2)(2a+5)=0`Rewrite the expressions`(6a^2+11a-10)/(3a-2) -: (2a+5)` `=` `((3a-2)(2a+5))/(3a-2) -: (2a+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)`Hint
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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)`Hint
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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)`