Power and product rules with positive expone
Simplify.

Ask by Stanley Potter. in the United States

Jan 04,2025

Question

Power and product rules with positive expone Simplify. $$ \left(-y^{4}\right)^{3}\left(-3 x y^{3} z^{3}\right)^{2} $$

UpStudy AI · Accepted Answer

To simplify the expression $$ \left(-y^{4}\right)^{3}\left(-3 x y^{3} z^{3}\right)^{2} $$, we'll use the **power of a product rule**, which states that $$(ab)^n = a^n b^n$$, and apply it step-by-step.

### Step 1: Apply the Power to Each Factor

1. **First Term: $$ \left(-y^{4}\right)^{3} $$**
   $$
   \left(-y^{4}\right)^{3} = (-1)^3 \cdot \left(y^{4}\right)^3 = -1 \cdot y^{12} = -y^{12}
   $$

2. **Second Term: $$ \left(-3 x y^{3} z^{3}\right)^{2} $$**
   $$
   \left(-3 x y^{3} z^{3}\right)^{2} = (-1)^2 \cdot 3^2 \cdot x^2 \cdot \left(y^{3}\right)^2 \cdot \left(z^{3}\right)^2 = 1 \cdot 9 \cdot x^2 \cdot y^{6} \cdot z^{6} = 9x^2 y^{6} z^{6}
   $$

### Step 2: Multiply the Simplified Terms

Now, multiply the results from both terms:
$$
(-y^{12}) \cdot (9x^2 y^{6} z^{6}) = -9x^2 y^{18} z^{6}
$$
*(Note that $$ y^{12} \cdot y^{6} = y^{18} $$)*

### Final Simplified Expression

$$
-9x^{2} y^{18} z^{6}
$$

**Answer:** $$-9\,x^{2}\,y^{18}\,z^{6}$$
$$-9x^{2}y^{18}z^{6}$$

Power and product rules with positive expone
Simplify.

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