# Solve by Factoring x^(-2/3)+x^(-1/3)-12=0

Rewrite as .
Let . Substitute for all occurrences of .
Factor using the AC method.
Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .
Write the factored form using these integers.
Replace all occurrences of with .
Simplify.
Rewrite the expression using the negative exponent rule .
Rewrite the expression using the negative exponent rule .
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
To write as a fraction with a common denominator, multiply by .
Combine the numerators over the common denominator.
Factor each term.
Multiply and .
Multiply by by adding the exponents.
Use the power rule to combine exponents.
Combine the numerators over the common denominator.
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.
Since contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part then find LCM for the variable part .
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.
The number is not a prime number because it only has one positive factor, which is itself.
Not prime
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either number.
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either term.
Multiply each term by and simplify.
Multiply each term in by in order to remove all the denominators from the equation.
Simplify .
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Expand using the FOIL Method.
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
Simplify each term.
Multiply by .
Multiply by .
Multiply by .
Rewrite using the commutative property of multiplication.
Multiply by by adding the exponents.
Use the power rule to combine exponents.
Combine the numerators over the common denominator.
Multiply by .
Subtract from .
Multiply by .
Solve the equation.
Find a common factor that is present in each term.
Substitute for .
Solve for .
Factor by grouping.
Reorder terms.
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
Multiply by .
Rewrite as plus
Apply the distributive property.
Factor out the greatest common factor from each group.
Group the first two terms and the last two terms.
Factor out the greatest common factor (GCF) from each group.
Factor the polynomial by factoring out the greatest common factor, .
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Set the first factor equal to and solve.
Set the first factor equal to .
Add to both sides of the equation.
Divide each term by and simplify.
Divide each term in by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Move the negative in front of the fraction.
Set the next factor equal to and solve.
Set the next factor equal to .
Add to both sides of the equation.
Divide each term by and simplify.
Divide each term in by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
The final solution is all the values that make true.
Substitute for .
Solve for for .
Raise each side of the equation to the power to eliminate the fractional exponent on the left side.
Simplify .
Use the power rule to distribute the exponent.
Apply the product rule to .
Apply the product rule to .
Raise to the power of .
One to any power is one.
Raise to the power of .
Solve for for .
Raise each side of the equation to the power to eliminate the fractional exponent on the left side.
Simplify .
Apply the product rule to .
One to any power is one.
Raise to the power of .
List all of the solutions.
Solve by Factoring x^(-2/3)+x^(-1/3)-12=0