Simplify (5z^-1+3)/(25z^-2-9)

Math
5z-1+325z-2-9
Simplify the numerator.
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Rewrite the expression using the negative exponent rule b-n=1bn.
51z+325z-2-9
Combine 5 and 1z.
5z+325z-2-9
To write 3 as a fraction with a common denominator, multiply by zz.
5z+3zz25z-2-9
Combine the numerators over the common denominator.
5+3zz25z-2-9
5+3zz25z-2-9
Simplify the denominator.
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Rewrite 25z-2 as (5z-1)2.
5+3zz(5z-1)2-9
Rewrite 9 as 32.
5+3zz(5z-1)2-32
Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b) where a=5z-1 and b=3.
5+3zz(5z-1+3)(5z-1-3)
Simplify.
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Rewrite the expression using the negative exponent rule b-n=1bn.
5+3zz(51z+3)(5z-1-3)
Combine 5 and 1z.
5+3zz(5z+3)(5z-1-3)
To write 3 as a fraction with a common denominator, multiply by zz.
5+3zz(5z+3zz)(5z-1-3)
Combine the numerators over the common denominator.
5+3zz5+3zz(5z-1-3)
Rewrite the expression using the negative exponent rule b-n=1bn.
5+3zz5+3zz(51z-3)
Combine 5 and 1z.
5+3zz5+3zz(5z-3)
To write -3 as a fraction with a common denominator, multiply by zz.
5+3zz5+3zz(5z-3⋅zz)
Combine -3 and zz.
5+3zz5+3zz(5z+-3zz)
Combine the numerators over the common denominator.
5+3zz5+3zz⋅5-3zz
5+3zz5+3zz⋅5-3zz
5+3zz5+3zz⋅5-3zz
Multiply 5+3zz and 5-3zz.
5+3zz(5+3z)(5-3z)z⋅z
Simplify the denominator.
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Raise z to the power of 1.
5+3zz(5+3z)(5-3z)z1z
Raise z to the power of 1.
5+3zz(5+3z)(5-3z)z1z1
Use the power rule aman=am+n to combine exponents.
5+3zz(5+3z)(5-3z)z1+1
Add 1 and 1.
5+3zz(5+3z)(5-3z)z2
5+3zz(5+3z)(5-3z)z2
Multiply the numerator by the reciprocal of the denominator.
5+3zz⋅z2(5+3z)(5-3z)
Cancel the common factor of 5+3z.
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Cancel the common factor.
5+3zz⋅z2(5+3z)(5-3z)
Rewrite the expression.
1z⋅z25-3z
1z⋅z25-3z
Cancel the common factor of z.
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Factor z out of z2.
1z⋅z⋅z5-3z
Cancel the common factor.
1z⋅z⋅z5-3z
Rewrite the expression.
z5-3z
z5-3z
Simplify (5z^-1+3)/(25z^-2-9)

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