Let's simplify the expression [tex]\(\frac{x^0 y^{-3}}{x^2 y^{-1}}\)[/tex] step-by-step.
1. Simplify the numerator and the denominator separately:
- The numerator is [tex]\(x^0 y^{-3}\)[/tex]:
- [tex]\(x^0 = 1\)[/tex] (Any number raised to the power of 0 is 1).
- So, the numerator becomes [tex]\(1 \cdot y^{-3} = y^{-3}\)[/tex].
- The denominator is [tex]\(x^2 y^{-1}\)[/tex]:
- It remains [tex]\(x^2 y^{-1}\)[/tex].
The expression now looks like this:
[tex]\[
\frac{y^{-3}}{x^2 y^{-1}}.
\][/tex]
2. Rewriting the expression using properties of exponents:
We can split the fraction into two separate fractions:
[tex]\[
\frac{y^{-3}}{x^2 y^{-1}} = \frac{y^{-3}}{x^2} \cdot \frac{1}{y^{-1}}.
\][/tex]
3. Simplify each part of the expression:
- The first part is [tex]\(\frac{y^{-3}}{x^2}\)[/tex]:
[tex]\[
\frac{y^{-3}}{x^2}.
\][/tex]
- The second part is [tex]\(\frac{1}{y^{-1}}\)[/tex]:
- We can rewrite [tex]\(y^{-1}\)[/tex] as [tex]\(\frac{1}{y^{-1}} = y^1 = y\)[/tex].
Now, we multiply these simplified parts together:
[tex]\[
\frac{y^{-3}}{x^2} \cdot y = \frac{y^{-3} \cdot y^1}{x^2}.
\][/tex]
4. Combine the exponents:
Using the properties of exponents, when you multiply terms with the same base, you add the exponents:
[tex]\[
y^{-3} \cdot y^1 = y^{-3+1} = y^{-2}.
\][/tex]
Thus, the expression simplifies to:
[tex]\[
\frac{y^{-2}}{x^2}.
\][/tex]
5. Express with positive exponents:
Since [tex]\(y^{-2}\)[/tex] can be rewritten with a positive exponent as [tex]\(\frac{1}{y^2}\)[/tex]:
[tex]\[
\frac{y^{-2}}{x^2} = \frac{1}{x^2 y^2}.
\][/tex]
Therefore, the simplified form of the given expression [tex]\(\frac{x^0 y^{-3}}{x^2 y^{-1}}\)[/tex] is [tex]\(\frac{1}{x^2 y^2}\)[/tex].
The correct answer is (B) [tex]\(\frac{1}{x^2 y^2}\)[/tex].
