Divide ((f^2-9)/(-f^3+6f^2))÷((-9f-27)/(f^2-12f+36))

Math
f2-9-f3+6f2÷-9f-27f2-12f+36
To divide by a fraction, multiply by its reciprocal.
f2-9-f3+6f2⋅f2-12f+36-9f-27
Simplify the numerator.
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Rewrite 9 as 32.
f2-32-f3+6f2⋅f2-12f+36-9f-27
Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b) where a=f and b=3.
(f+3)(f-3)-f3+6f2⋅f2-12f+36-9f-27
(f+3)(f-3)-f3+6f2⋅f2-12f+36-9f-27
Factor f2 out of -f3+6f2.
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Factor f2 out of -f3.
(f+3)(f-3)f2(-f)+6f2⋅f2-12f+36-9f-27
Factor f2 out of 6f2.
(f+3)(f-3)f2(-f)+f2⋅6⋅f2-12f+36-9f-27
Factor f2 out of f2(-f)+f2⋅6.
(f+3)(f-3)f2(-f+6)⋅f2-12f+36-9f-27
(f+3)(f-3)f2(-f+6)⋅f2-12f+36-9f-27
Factor using the perfect square rule.
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Rewrite 36 as 62.
(f+3)(f-3)f2(-f+6)⋅f2-12f+62-9f-27
Check the middle term by multiplying 2ab and compare this result with the middle term in the original expression.
2ab=2⋅f⋅-6
Simplify.
2ab=-12f
Factor using the perfect square trinomial rule a2-2ab+b2=(a-b)2, where a=f and b=-6.
(f+3)(f-3)f2(-f+6)⋅(f-6)2-9f-27
(f+3)(f-3)f2(-f+6)⋅(f-6)2-9f-27
Factor 9 out of -9f-27.
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Factor 9 out of -9f.
(f+3)(f-3)f2(-f+6)⋅(f-6)29(-f)-27
Factor 9 out of -27.
(f+3)(f-3)f2(-f+6)⋅(f-6)29(-f)+9(-3)
Factor 9 out of 9(-f)+9(-3).
(f+3)(f-3)f2(-f+6)⋅(f-6)29(-f-3)
(f+3)(f-3)f2(-f+6)⋅(f-6)29(-f-3)
Multiply (f+3)(f-3)f2(-f+6) and (f-6)29(-f-3).
(f+3)(f-3)(f-6)2f2(-f+6)(9(-f-3))
Cancel the common factor of f+3 and -f-3.
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Factor -1 out of f.
(-1(-f)+3)(f-3)(f-6)2f2(-f+6)(9(-f-3))
Rewrite 3 as -1(-3).
(-1(-f)-1(-3))(f-3)(f-6)2f2(-f+6)(9(-f-3))
Factor -1 out of -1(-f)-1(-3).
-1(-f-3)(f-3)(f-6)2f2(-f+6)(9(-f-3))
Cancel the common factor.
-1(-f-3)(f-3)(f-6)2f2(-f+6)⋅9(-f-3)
Rewrite the expression.
(-1(f-3))(f-6)2f2(-f+6)⋅9
(-1(f-3))(f-6)2f2(-f+6)⋅9
Cancel the common factor of (f-6)2 and -f+6.
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Factor -1 out of f.
-1(f-3)(-1(-f)-6)2f2(-f+6)⋅9
Rewrite -6 as -1(6).
-1(f-3)(-1(-f)-1(6))2f2(-f+6)⋅9
Factor -1 out of -1(-f)-1(6).
-1(f-3)(-1(-f+6))2f2(-f+6)⋅9
Apply the product rule to -1(-f+6).
-1(f-3)((-1)2(-f+6)2)f2(-f+6)⋅9
Raise -1 to the power of 2.
-1(f-3)(1(-f+6)2)f2(-f+6)⋅9
Multiply (-f+6)2 by 1.
-1(f-3)(-f+6)2f2(-f+6)⋅9
Factor -f+6 out of -1(f-3)(-f+6)2.
(-f+6)(-1(f-3)(-f+6))f2(-f+6)⋅9
Cancel the common factors.
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Factor -f+6 out of f2(-f+6)⋅9.
(-f+6)(-1(f-3)(-f+6))(-f+6)(f2⋅9)
Cancel the common factor.
(-f+6)(-1(f-3)(-f+6))(-f+6)(f2⋅9)
Rewrite the expression.
-1(f-3)(-f+6)f2⋅9
-1(f-3)(-f+6)f2⋅9
-1(f-3)(-f+6)f2⋅9
Move 9 to the left of f2.
-1(f-3)(-f+6)9⋅f2
Move the negative in front of the fraction.
-(f-3)(-f+6)9⋅f2
Factor -1 out of -f.
-(f-3)(-(f)+6)9f2
Rewrite 6 as -1(-6).
-(f-3)(-(f)-1(-6))9f2
Factor -1 out of -(f)-1(-6).
-(f-3)(-(f-6))9f2
Rewrite -(f-6) as -1(f-6).
-(f-3)(-1(f-6))9f2
Move the negative in front of the fraction.
–(f-3)(f-6)9f2
Multiply -1 by -1.
1(f-3)(f-6)9f2
Multiply (f-3)(f-6)9f2 by 1.
(f-3)(f-6)9f2
Expand (f-3)(f-6) using the FOIL Method.
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Apply the distributive property.
f(f-6)-3(f-6)9f2
Apply the distributive property.
f⋅f+f⋅-6-3(f-6)9f2
Apply the distributive property.
f⋅f+f⋅-6-3f-3⋅-69f2
f⋅f+f⋅-6-3f-3⋅-69f2
Simplify and combine like terms.
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Simplify each term.
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Multiply f by f.
f2+f⋅-6-3f-3⋅-69f2
Move -6 to the left of f.
f2-6⋅f-3f-3⋅-69f2
Multiply -3 by -6.
f2-6f-3f+189f2
f2-6f-3f+189f2
Subtract 3f from -6f.
f2-9f+189f2
f2-9f+189f2
Split the fraction f2-9f+189f2 into two fractions.
f2-9f9f2+189f2
Split the fraction f2-9f9f2 into two fractions.
f29f2+-9f9f2+189f2
Cancel the common factor of f2.
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Cancel the common factor.
f29f2+-9f9f2+189f2
Rewrite the expression.
19+-9f9f2+189f2
19+-9f9f2+189f2
Cancel the common factor of -9 and 9.
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Factor 9 out of -9f.
19+9(-f)9f2+189f2
Cancel the common factors.
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Factor 9 out of 9f2.
19+9(-f)9(f2)+189f2
Cancel the common factor.
19+9(-f)9f2+189f2
Rewrite the expression.
19+-ff2+189f2
19+-ff2+189f2
19+-ff2+189f2
Cancel the common factor of f and f2.
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Factor f out of -f.
19+f⋅-1f2+189f2
Cancel the common factors.
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Factor f out of f2.
19+f⋅-1f⋅f+189f2
Cancel the common factor.
19+f⋅-1f⋅f+189f2
Rewrite the expression.
19+-1f+189f2
19+-1f+189f2
19+-1f+189f2
Move the negative in front of the fraction.
19-1f+189f2
Cancel the common factor of 18 and 9.
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Factor 9 out of 18.
19-1f+9⋅29f2
Cancel the common factors.
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Factor 9 out of 9f2.
19-1f+9⋅29(f2)
Cancel the common factor.
19-1f+9⋅29f2
Rewrite the expression.
19-1f+2f2
19-1f+2f2
19-1f+2f2
Divide ((f^2-9)/(-f^3+6f^2))÷((-9f-27)/(f^2-12f+36))

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