Simplify (2x^0 / (4x^{-2} - 2y^4))^2. Write your answer using only positive exponents.
Understand the Problem
The question is asking to simplify the given expression involving rational exponents, specifically \((\frac{2x^0}{4x^{-2}-2y^4})^2\), and to present the final answer using only positive exponents.
Answer
$$ \frac{4x^4}{(4 - 2y^4 x^2)^2} $$
Answer for screen readers
The simplified expression is: $$ \frac{4x^4}{(4 - 2y^4 x^2)^2} $$
Steps to Solve
- Simplify the numerator
The numerator is $2x^0$. Recall that $x^0 = 1$, so we have: $$ 2x^0 = 2 \times 1 = 2 $$
- Simplify the denominator
The denominator is $4x^{-2} - 2y^4$. To simplify $4x^{-2}$, recall that $x^{-2} = \frac{1}{x^2}$. Thus, $$ 4x^{-2} = \frac{4}{x^2} $$
So the denominator becomes: $$ \frac{4}{x^2} - 2y^4 $$
- Combine the terms in the denominator
To combine the terms, we need a common denominator. The common denominator is $x^2$: $$ \frac{4}{x^2} - 2y^4 \cdot \frac{x^2}{x^2} = \frac{4 - 2y^4 x^2}{x^2} $$
- Write the expression with the new denominator
Now rewrite the original expression using the simplified numerator and denominator: $$ \left( \frac{2}{\frac{4 - 2y^4 x^2}{x^2}} \right)^2 $$
This is equivalent to: $$ \left( \frac{2x^2}{4 - 2y^4 x^2} \right)^2 $$
- Square the entire expression
Now square the numerator and denominator: $$ \frac{(2x^2)^2}{(4 - 2y^4 x^2)^2} = \frac{4x^4}{(4 - 2y^4 x^2)^2} $$
The simplified expression is: $$ \frac{4x^4}{(4 - 2y^4 x^2)^2} $$
More Information
This expression represents a rational function where the numerator is a polynomial and the denominator is a squared polynomial. It's important to always express the answer using positive exponents.
Tips
- Forgetting that $x^0 = 1$, leading to incorrect simplification of the numerator.
- Not recognizing the need for a common denominator when combining terms in the denominator.
- Failing to square both the numerator and denominator correctly during the final step.
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