# Solve for z z/3+3/z=z/12+12/z z3+3z=z12+12z
Find the LCD of the terms in the equation.
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
3,z,12,z
Since 3,z,12,z contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part 3,1,12,1 then find LCM for the variable part z1,z1.
The LCM is the smallest positive number that all of the numbers divide into evenly.
1. List the prime factors of each number.
2. Multiply each factor the greatest number of times it occurs in either number.
Since 3 has no factors besides 1 and 3.
3 is a prime number
The number 1 is not a prime number because it only has one positive factor, which is itself.
Not prime
The prime factors for 12 are 2⋅2⋅3.
12 has factors of 2 and 6.
2⋅6
6 has factors of 2 and 3.
2⋅2⋅3
2⋅2⋅3
The number 1 is not a prime number because it only has one positive factor, which is itself.
Not prime
The LCM of 3,1,12,1 is the result of multiplying all prime factors the greatest number of times they occur in either number.
2⋅2⋅3
The LCM of 3,1,12,1 is 2⋅2⋅3=12.
Multiply 2 by 2.
4⋅3
Multiply 4 by 3.
12
12
The factor for z1 is z itself.
z1=z
z occurs 1 time.
The LCM of z1,z1 is the result of multiplying all prime factors the greatest number of times they occur in either term.
z
The LCM for 3,z,12,z is the numeric part 12 multiplied by the variable part.
12z
12z
Multiply each term by 12z and simplify.
Multiply each term in z3+3z=z12+12z by 12z in order to remove all the denominators from the equation.
z3⋅(12z)+3z⋅(12z)=z12⋅(12z)+12z⋅(12z)
Simplify each term.
Rewrite using the commutative property of multiplication.
12z3z+3z⋅(12z)=z12⋅(12z)+12z⋅(12z)
Cancel the common factor of 3.
Factor 3 out of 12.
3(4)z3z+3z⋅(12z)=z12⋅(12z)+12z⋅(12z)
Cancel the common factor.
3⋅4z3z+3z⋅(12z)=z12⋅(12z)+12z⋅(12z)
Rewrite the expression.
4z⋅z+3z⋅(12z)=z12⋅(12z)+12z⋅(12z)
4z⋅z+3z⋅(12z)=z12⋅(12z)+12z⋅(12z)
Multiply z by z by adding the exponents.
Move z.
4(z⋅z)+3z⋅(12z)=z12⋅(12z)+12z⋅(12z)
Multiply z by z.
4z2+3z⋅(12z)=z12⋅(12z)+12z⋅(12z)
4z2+3z⋅(12z)=z12⋅(12z)+12z⋅(12z)
Rewrite using the commutative property of multiplication.
4z2+123zz=z12⋅(12z)+12z⋅(12z)
Multiply 123z.
Combine 12 and 3z.
4z2+12⋅3zz=z12⋅(12z)+12z⋅(12z)
Multiply 12 by 3.
4z2+36zz=z12⋅(12z)+12z⋅(12z)
4z2+36zz=z12⋅(12z)+12z⋅(12z)
Cancel the common factor of z.
Cancel the common factor.
4z2+36zz=z12⋅(12z)+12z⋅(12z)
Rewrite the expression.
4z2+36=z12⋅(12z)+12z⋅(12z)
4z2+36=z12⋅(12z)+12z⋅(12z)
4z2+36=z12⋅(12z)+12z⋅(12z)
Simplify each term.
Rewrite using the commutative property of multiplication.
4z2+36=12z12z+12z⋅(12z)
Cancel the common factor of 12.
Cancel the common factor.
4z2+36=12z12z+12z⋅(12z)
Rewrite the expression.
4z2+36=z⋅z+12z⋅(12z)
4z2+36=z⋅z+12z⋅(12z)
Multiply z by z.
4z2+36=z2+12z⋅(12z)
Rewrite using the commutative property of multiplication.
4z2+36=z2+1212zz
Multiply 1212z.
Combine 12 and 12z.
4z2+36=z2+12⋅12zz
Multiply 12 by 12.
4z2+36=z2+144zz
4z2+36=z2+144zz
Cancel the common factor of z.
Cancel the common factor.
4z2+36=z2+144zz
Rewrite the expression.
4z2+36=z2+144
4z2+36=z2+144
4z2+36=z2+144
4z2+36=z2+144
Solve the equation.
Move all terms containing z to the left side of the equation.
Subtract z2 from both sides of the equation.
4z2+36-z2=144
Subtract z2 from 4z2.
3z2+36=144
3z2+36=144
Move all terms not containing z to the right side of the equation.
Subtract 36 from both sides of the equation.
3z2=144-36
Subtract 36 from 144.
3z2=108
3z2=108
Divide each term by 3 and simplify.
Divide each term in 3z2=108 by 3.
3z23=1083
Cancel the common factor of 3.
Cancel the common factor.
3z23=1083
Divide z2 by 1.
z2=1083
z2=1083
Divide 108 by 3.
z2=36
z2=36
Take the square root of both sides of the equation to eliminate the exponent on the left side.
z=±36
The complete solution is the result of both the positive and negative portions of the solution.
Simplify the right side of the equation.
Rewrite 36 as 62.
z=±62
Pull terms out from under the radical, assuming positive real numbers.
z=±6
z=±6
The complete solution is the result of both the positive and negative portions of the solution.
First, use the positive value of the ± to find the first solution.
z=6
Next, use the negative value of the ± to find the second solution.
z=-6
The complete solution is the result of both the positive and negative portions of the solution.
z=6,-6
z=6,-6
z=6,-6
z=6,-6
Solve for z z/3+3/z=z/12+12/z

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