Laws Exponents 4C319D

1. **Problem:** Simplify $(3x^4y^2) \cdot (5x^3y^5)$ completely. **Step 1:** Use the product rule for exponents: $a^m \cdot a^n = a^{m+n}$. **Step 2:** Multiply coefficients: $3 \times 5 = 15$. **Step 3:** Add exponents for $x$: $4 + 3 = 7$. **Step 4:** Add exponents for $y$: $2 + 5 = 7$. **Answer:** $15x^7y^7$ 2. **Problem:** Simplify $(2x^3y^2)^4$ completely. **Step 1:** Use the power of a product rule: $(abc)^n = a^n b^n c^n$. **Step 2:** Raise each factor to the 4th power: $$2^4 = 16$$ $$x^{3 \times 4} = x^{12}$$ $$y^{2 \times 4} = y^8$$ **Answer:** $16x^{12}y^8$ 3. **Problem:** Simplify and write with positive exponents: $\frac{18a^5b^3}{6a^2b^7}$. **Step 1:** Simplify coefficients: $\frac{18}{6} = \cancel{3}$ **Step 2:** Subtract exponents for $a$: $5 - 2 = 3$. **Step 3:** Subtract exponents for $b$: $3 - 7 = -4$. **Step 4:** Rewrite $b^{-4}$ as $\frac{1}{b^4}$ to keep positive exponents. **Answer:** $3a^3 \frac{1}{b^4} = \frac{3a^3}{b^4}$ 4. **Problem:** Evaluate and simplify $(4x^{-3}y^2)^0$. **Step 1:** Any nonzero expression raised to the zero power equals 1. **Answer:** $1$