- The objective is to identify which of the given expressions matches \(\frac{y^2}{x^2}\) after removing negative exponents.
- We simplify each option: (a) becomes \(x\), (b) becomes \(\frac{x^4}{y^8}\), (c) remains \(\frac{x^4}{y^4}\), and (d) remains \(\frac{x^4}{y^2+y^6}\).
- After testing each case, we find that none of the expressions are equivalent to the target expression.
- \(\boxed{\text{None of the options are correct}}\)
Explanation
- Analysis of the problem
The goal is to determine which of the given expressions is equivalent to \(\frac{y^2}{x^2}\), assuming \(x
eq 0\) and \(y
eq 0\). We are looking for an expression that, when simplified, results in \(\frac{y^2}{x^2}\). We will analyze each option individually by simplifying them using algebraic rules.
- Evaluation of options
We analyze each option to see if it simplifies to \(\frac{y^2}{x^2}\):
AFor \(\frac{x^2 y^2}{x y^2}\), we can cancel the common terms. Since \(y^2 / y^2 = 1\) and \(x^2 / x = x\), this simplifies to:
\[\frac{x^2 y^2}{x y^2} = x \cdot \left(\frac{y^2}{y^2}\right) = x \cdot 1 = x\]
This is not equivalent to \(\frac{y^2}{x^2}\).
BFor \(\frac{x^4}{y^2 y^6}\), we use the rule for exponents \(y^a \cdot y^b = y^{a+b}\):
\[\frac{x^4}{y^2 \cdot y^6} = \frac{x^4}{y^{2+6}} = \frac{x^4}{y^8}\]
This is not equivalent to \(\frac{y^2}{x^2}\).
CFor \(\frac{x^4}{y^4}\), this is \(\left(\frac{x}{y}\right)^4\), which is not equivalent to \(\frac{y^2}{x^2}\).
DFor \(\frac{x^4}{y^2+y^6}\), this expression cannot be simplified by canceling terms because the denominator is a sum, not a product. It is clearly not equal to \(\frac{y^2}{x^2}\).
- Conclusion
Based on our step-by-step evaluation of the provided options, none of them simplify to the target expression \(\frac{y^2}{x^2}\). Therefore, there may be a discrepancy or typo in the provided options within the question. Within the scope of the given choices, none are correct.
Examples
In real-world engineering or physics, simplified algebraic expressions are crucial for calculating values like stress, strain, or power. For example, if you are calculating the power dissipated by a resistor, \(P = I^2 R\), and your formula has been derived using negative exponents, you must correctly convert those back into a standard form to calculate the actual power. If the derived formula does not match the standard simplified form, it could lead to incorrect safety ratings for electrical components. Always double-check your algebraic simplifications to ensure they match established physical formulas.