# Simplify ((z^2-64)/(z^2-14z+49))÷((6z-48)/(z^2-6z-7)) z2-64z2-14z+49÷6z-48z2-6z-7
To divide by a fraction, multiply by its reciprocal.
z2-64z2-14z+49⋅z2-6z-76z-48
Simplify the numerator.
Rewrite 64 as 82.
z2-82z2-14z+49⋅z2-6z-76z-48
Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b) where a=z and b=8.
(z+8)(z-8)z2-14z+49⋅z2-6z-76z-48
(z+8)(z-8)z2-14z+49⋅z2-6z-76z-48
Factor using the perfect square rule.
Rewrite 49 as 72.
(z+8)(z-8)z2-14z+72⋅z2-6z-76z-48
Check the middle term by multiplying 2ab and compare this result with the middle term in the original expression.
2ab=2⋅z⋅-7
Simplify.
2ab=-14z
Factor using the perfect square trinomial rule a2-2ab+b2=(a-b)2, where a=z and b=-7.
(z+8)(z-8)(z-7)2⋅z2-6z-76z-48
(z+8)(z-8)(z-7)2⋅z2-6z-76z-48
Factor z2-6z-7 using the AC method.
Consider the form x2+bx+c. Find a pair of integers whose product is c and whose sum is b. In this case, whose product is -7 and whose sum is -6.
-7,1
Write the factored form using these integers.
(z+8)(z-8)(z-7)2⋅(z-7)(z+1)6z-48
(z+8)(z-8)(z-7)2⋅(z-7)(z+1)6z-48
Simplify terms.
Factor 6 out of 6z-48.
Factor 6 out of 6z.
(z+8)(z-8)(z-7)2⋅(z-7)(z+1)6(z)-48
Factor 6 out of -48.
(z+8)(z-8)(z-7)2⋅(z-7)(z+1)6z+6⋅-8
Factor 6 out of 6z+6⋅-8.
(z+8)(z-8)(z-7)2⋅(z-7)(z+1)6(z-8)
(z+8)(z-8)(z-7)2⋅(z-7)(z+1)6(z-8)
Cancel the common factor of z-8.
Factor z-8 out of (z+8)(z-8).
(z-8)(z+8)(z-7)2⋅(z-7)(z+1)6(z-8)
Factor z-8 out of 6(z-8).
(z-8)(z+8)(z-7)2⋅(z-7)(z+1)(z-8)⋅6
Cancel the common factor.
(z-8)(z+8)(z-7)2⋅(z-7)(z+1)(z-8)⋅6
Rewrite the expression.
z+8(z-7)2⋅(z-7)(z+1)6
z+8(z-7)2⋅(z-7)(z+1)6
Cancel the common factor of z-7.
Factor z-7 out of (z-7)2.
z+8(z-7)(z-7)⋅(z-7)(z+1)6
Cancel the common factor.
z+8(z-7)(z-7)⋅(z-7)(z+1)6
Rewrite the expression.
z+8z-7⋅z+16
z+8z-7⋅z+16
Multiply z+8z-7 and z+16.
(z+8)(z+1)(z-7)⋅6
Move 6 to the left of z-7.
(z+8)(z+1)6(z-7)
(z+8)(z+1)6(z-7)
Simplify ((z^2-64)/(z^2-14z+49))÷((6z-48)/(z^2-6z-7))

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