2.2. Simplify the following: $$ 2.2 .1 \quad 4 x^{2} y^{-4} imes\left(\frac{2 y^{3}}{x^{3}} ight)^{3} $$

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2.2. Simplify the following: $$ 2.2 .1 \quad 4 x^{2} y^{-4} 	imes\left(\frac{2 y^{3}}{x^{3}}ight)^{3} $$

UpStudy AI · Accepted Answer

We begin with the expression:
$$
4x^2 y^{-4} \times \left(\frac{2y^3}{x^3}\right)^3.
$$

**Step 1. Expand the exponent.**

Apply the exponent $$3$$ to both numerator and denominator:
$$
\left(\frac{2y^3}{x^3}\right)^3 = \frac{(2y^3)^3}{(x^3)^3}.
$$
This becomes:
$$
\frac{2^3 \, (y^3)^3}{x^{9}} = \frac{8 \, y^9}{x^9}.
$$

**Step 2. Multiply the expressions.**

The original expression now is:
$$
4x^2 y^{-4} \times \frac{8y^9}{x^9}.
$$
Multiply the coefficients, the $$x$$-terms, and the $$y$$-terms separately:
$$
\text{Coefficient: } 4 \times 8 = 32.
$$
$$
\text{$$x$$-terms: } x^2 \times \frac{1}{x^9} = x^{2-9} = x^{-7}.
$$
$$
\text{$$y$$-terms: } y^{-4} \times y^9 = y^{-4+9} = y^5.
$$
Thus, the expression simplifies to:
$$
32\, x^{-7}\, y^5.
$$

**Step 3. Write the final answer using positive exponents.**

Since $$x^{-7} = \frac{1}{x^7}$$, the final simplified form is:
$$
\frac{32\, y^5}{x^7}.
$$
$$ \frac{32y^5}{x^7} $$

2.2. Simplify the following: \( 2.2 .1 \quad 4 x^{2} y^{-4} \times\left(\frac{2 y^{3}}{x^{3}}\right)^{3} \)

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