Finite Math Examples

Find the Domain ( cube root of x- ninth root of x)/(3 cube root of x+ ninth root of x)
Step 1
Set the denominator in equal to to find where the expression is undefined.
Step 2
Solve for .
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Step 2.1
Subtract from both sides of the equation.
Step 2.2
To remove the radical on the left side of the equation, cube both sides of the equation.
Step 2.3
Simplify each side of the equation.
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Step 2.3.1
Use to rewrite as .
Step 2.3.2
Simplify the left side.
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Step 2.3.2.1
Simplify .
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Step 2.3.2.1.1
Apply the product rule to .
Step 2.3.2.1.2
Raise to the power of .
Step 2.3.2.1.3
Multiply the exponents in .
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Step 2.3.2.1.3.1
Apply the power rule and multiply exponents, .
Step 2.3.2.1.3.2
Cancel the common factor of .
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Step 2.3.2.1.3.2.1
Cancel the common factor.
Step 2.3.2.1.3.2.2
Rewrite the expression.
Step 2.3.2.1.4
Simplify.
Step 2.3.3
Simplify the right side.
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Step 2.3.3.1
Simplify .
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Step 2.3.3.1.1
Simplify the expression.
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Step 2.3.3.1.1.1
Apply the product rule to .
Step 2.3.3.1.1.2
Raise to the power of .
Step 2.3.3.1.2
Rewrite as .
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Step 2.3.3.1.2.1
Use to rewrite as .
Step 2.3.3.1.2.2
Apply the power rule and multiply exponents, .
Step 2.3.3.1.2.3
Combine and .
Step 2.3.3.1.2.4
Cancel the common factor of and .
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Step 2.3.3.1.2.4.1
Factor out of .
Step 2.3.3.1.2.4.2
Cancel the common factors.
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Step 2.3.3.1.2.4.2.1
Factor out of .
Step 2.3.3.1.2.4.2.2
Cancel the common factor.
Step 2.3.3.1.2.4.2.3
Rewrite the expression.
Step 2.3.3.1.2.5
Rewrite as .
Step 2.4
Rewrite the equation as .
Step 2.5
To remove the radical on the left side of the equation, cube both sides of the equation.
Step 2.6
Simplify each side of the equation.
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Step 2.6.1
Use to rewrite as .
Step 2.6.2
Simplify the left side.
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Step 2.6.2.1
Simplify .
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Step 2.6.2.1.1
Apply the product rule to .
Step 2.6.2.1.2
Raise to the power of .
Step 2.6.2.1.3
Multiply the exponents in .
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Step 2.6.2.1.3.1
Apply the power rule and multiply exponents, .
Step 2.6.2.1.3.2
Cancel the common factor of .
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Step 2.6.2.1.3.2.1
Cancel the common factor.
Step 2.6.2.1.3.2.2
Rewrite the expression.
Step 2.6.2.1.4
Simplify.
Step 2.6.3
Simplify the right side.
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Step 2.6.3.1
Simplify .
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Step 2.6.3.1.1
Apply the product rule to .
Step 2.6.3.1.2
Raise to the power of .
Step 2.7
Solve for .
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Step 2.7.1
Subtract from both sides of the equation.
Step 2.7.2
Factor out of .
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Step 2.7.2.1
Reorder and .
Step 2.7.2.2
Factor out of .
Step 2.7.2.3
Factor out of .
Step 2.7.2.4
Factor out of .
Step 2.7.3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 2.7.4
Set equal to .
Step 2.7.5
Set equal to and solve for .
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Step 2.7.5.1
Set equal to .
Step 2.7.5.2
Solve for .
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Step 2.7.5.2.1
Subtract from both sides of the equation.
Step 2.7.5.2.2
Divide each term in by and simplify.
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Step 2.7.5.2.2.1
Divide each term in by .
Step 2.7.5.2.2.2
Simplify the left side.
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Step 2.7.5.2.2.2.1
Cancel the common factor of .
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Step 2.7.5.2.2.2.1.1
Cancel the common factor.
Step 2.7.5.2.2.2.1.2
Divide by .
Step 2.7.5.2.2.3
Simplify the right side.
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Step 2.7.5.2.2.3.1
Move the negative in front of the fraction.
Step 2.7.5.2.3
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 2.7.5.2.4
Simplify .
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Step 2.7.5.2.4.1
Rewrite as .
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Step 2.7.5.2.4.1.1
Rewrite as .
Step 2.7.5.2.4.1.2
Factor the perfect power out of .
Step 2.7.5.2.4.1.3
Factor the perfect power out of .
Step 2.7.5.2.4.1.4
Rearrange the fraction .
Step 2.7.5.2.4.1.5
Rewrite as .
Step 2.7.5.2.4.2
Pull terms out from under the radical.
Step 2.7.5.2.4.3
Rewrite as .
Step 2.7.5.2.4.4
Any root of is .
Step 2.7.5.2.4.5
Multiply by .
Step 2.7.5.2.4.6
Combine and simplify the denominator.
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Step 2.7.5.2.4.6.1
Multiply by .
Step 2.7.5.2.4.6.2
Raise to the power of .
Step 2.7.5.2.4.6.3
Raise to the power of .
Step 2.7.5.2.4.6.4
Use the power rule to combine exponents.
Step 2.7.5.2.4.6.5
Add and .
Step 2.7.5.2.4.6.6
Rewrite as .
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Step 2.7.5.2.4.6.6.1
Use to rewrite as .
Step 2.7.5.2.4.6.6.2
Apply the power rule and multiply exponents, .
Step 2.7.5.2.4.6.6.3
Combine and .
Step 2.7.5.2.4.6.6.4
Cancel the common factor of .
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Step 2.7.5.2.4.6.6.4.1
Cancel the common factor.
Step 2.7.5.2.4.6.6.4.2
Rewrite the expression.
Step 2.7.5.2.4.6.6.5
Evaluate the exponent.
Step 2.7.5.2.4.7
Multiply .
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Step 2.7.5.2.4.7.1
Multiply by .
Step 2.7.5.2.4.7.2
Multiply by .
Step 2.7.5.2.5
The complete solution is the result of both the positive and negative portions of the solution.
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Step 2.7.5.2.5.1
First, use the positive value of the to find the first solution.
Step 2.7.5.2.5.2
Next, use the negative value of the to find the second solution.
Step 2.7.5.2.5.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 2.7.6
The final solution is all the values that make true.
Step 3
The domain is all values of that make the expression defined.
Interval Notation:
Set-Builder Notation:
Step 4