4. $$ \left(\frac{5 x^{0}}{10 x^{-3} y^{2}}\right)^{2} $$

Question

UpStudy AI · Accepted Answer

1. Simplify the expression inside the parentheses. Note that $$ x^0 = 1 $$, so
   $$
   \frac{5x^{0}}{10x^{-3}y^{2}} = \frac{5}{10x^{-3}y^{2}}.
   $$
2. Simplify the numerical coefficient:
   $$
   \frac{5}{10} = \frac{1}{2}.
   $$
   Thus, the expression becomes
   $$
   \frac{1}{2} \cdot \frac{1}{x^{-3}y^{2}}.
   $$
3. Recall that $$ \frac{1}{x^{-3}} = x^{3} $$. Therefore,
   $$
   \frac{1}{x^{-3}y^{2}} = x^{3} \cdot \frac{1}{y^{2}} = \frac{x^{3}}{y^{2}}.
   $$
   Combining this with the previous result, we have
   $$
   \frac{1}{2} \cdot \frac{x^{3}}{y^{2}} = \frac{x^{3}}{2y^{2}}.
   $$
4. Raise the simplified expression to the power of 2:
   $$
   \left(\frac{x^{3}}{2y^{2}}\right)^{2}.
   $$
5. Apply the exponent to both the numerator and the denominator:
   $$
   \frac{(x^{3})^{2}}{(2y^{2})^{2}} = \frac{x^{3 \cdot 2}}{2^{2} \cdot y^{2 \cdot 2}} = \frac{x^{6}}{4y^{4}}.
   $$

The simplified expression is:
$$
\frac{x^{6}}{4y^{4}}.
$$
The simplified expression is $$ \frac{x^{6}}{4y^{4}} $$.

4. \( \left(\frac{5 x^{0}}{10 x^{-3} y^{2}}\right)^{2} \)

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