Simplifying Radicals

Category: Algebra

"Published in Newark, California, USA"

 Simplify

Solution:

Consider the given equation above

We can rewrite the above equation as follows

 Next, we need to eliminate the radical sign at the denominator by rationalization of the denominator. The exponent of 3 is 2 and the exponent of y is 3. We need to multiply both the numerator and denominator by 3²y as follows

Since the exponent of 3 is 2, the exponent of x is 1, and the exponent of y is 1, then we cannot simplify further because the index of the radical is 4. Therefore, the final answer is

or

Indeterminate Form - Infinity Minus Infinity, 2

Category: Differential Calculus, Trigonometry

"Published in Suisun City, California, USA"

Evaluate

Solution:

Consider the given equation above

Substitute the value of x to the above equation, we have

Since the final answer is ∞ - ∞, then it is Indeterminate Form which is not accepted as a final answer in Mathematics. We cannot use the L'Hopital's Rule because L'Hopital's Rule is applicable only for Indeterminate Forms like 0/0 and ∞/∞. The above equation must be rewritten and expressed into another form until the Indeterminate Form is 0/0 or ∞/∞. Let's rewrite the above equation as follows

Substitute the value of x to the above equation, we have

Since the Indeterminate Form is 0/0, then we can apply the L'Hopital's Rule as follows

Substitute the value of x to the above equation, we have

Therefore, 

Derivative - Algebraic, Rational Function

Category: Differential Calculus, Algebra

"Published in Suisun City, California, USA"

Find the derivative for

Solution:

Consider the given equation above

Take the derivative of the given equation with respect to y, we have

Apply the derivative by power at the right side of the equation

Next, apply the derivative of rational function at the right side of the equation