To solve the expression [tex]\(\frac{2 x^0 y^2 \cdot 2 y^3}{\left(2 x^{-2}\right)^2}\)[/tex], we’ll proceed step-by-step:
### Step 1: Simplify the numerator
Consider the numerator [tex]\(2 x^0 y^2 \cdot 2 y^3\)[/tex]:
1. Multiply the constants: [tex]\(2 \cdot 2 = 4\)[/tex].
2. Next, simplify the variable parts. Since [tex]\(x^0 = 1\)[/tex], the [tex]\(x^0\)[/tex] term can be omitted, simplifying it to:
[tex]\[
4 y^2 y^3
\][/tex]
3. Using the properties of exponents, combine [tex]\(y^2 \cdot y^3\)[/tex]:
[tex]\[
y^{2+3} = y^5
\][/tex]
Thus, the simplified numerator is:
[tex]\[
4 y^5
\][/tex]
### Step 2: Simplify the denominator
Consider the denominator [tex]\((2 x^{-2})^2\)[/tex]:
1. Distribute the exponent across the product inside the parentheses:
[tex]\[
(2 x^{-2})^2 = 2^2 \cdot (x^{-2})^2
\][/tex]
2. Calculate [tex]\(2^2\)[/tex]:
[tex]\[
2^2 = 4
\][/tex]
3. Calculate [tex]\((x^{-2})^2\)[/tex] using the properties of exponents:
[tex]\[
(x^{-2})^2 = x^{-2 \cdot 2} = x^{-4}
\][/tex]
Thus, the simplified denominator is:
[tex]\[
4 x^{-4}
\][/tex]
### Step 3: Combine the simplified numerator and denominator
1. Express the entire fraction:
[tex]\[
\frac{4 y^5}{4 x^{-4}}
\][/tex]
2. The 4s in the numerator and denominator cancel each other out.
[tex]\[
\frac{y^5}{x^{-4}} = y^5 \cdot x^4 \quad \text{(because dividing by \(x^{-4}\) is the same as multiplying by \(x^4\))}
\][/tex]
### Conclusion
Thus, the simplified expression is:
[tex]\[
x^4 y^5
\][/tex]
