Question

1. $4xy^{3} \cdot 2x^{3}$
2. $(3n)^{4}$
3. $(2n^{2})^{2}$
4. $\frac{4xy^{4}}{4x^{3}y^{2}}$
5. $\frac{x^{3}y^{4}}{4y^{4}}$
6. $-\frac{2r^{4}}{4r^{2}}$
7. $\frac{-2b^{4}}{-b^{3}}$

Explanation:

Step1: Multiply coefficients, combine $x$ terms

$4 \cdot 2 \cdot x^{1+3} y^3 = 8x^4y^3$

Step2: Distribute exponent to factors

$3^4 n^4 = 81n^4$

Step3: Distribute exponent to factors

$2^2 (n^2)^2 = 4n^{4}$

Step4: Simplify coefficients, subtract exponents

$\frac{4}{4} x^{1-3} y^{4-2} = x^{-2}y^2 = \frac{y^2}{x^2}$

Step5: Cancel $y^4$, simplify coefficient

$\frac{x^3}{4} \cdot \frac{y^4}{y^4} = \frac{x^3}{4}$

Step6: Simplify coefficients, subtract exponents

$-\frac{2}{4} r^{4-2} = -\frac{1}{2}r^2$

Step7: Cancel negatives, subtract exponents

$\frac{-2}{-1} b^{4-3} = 2b$

Answer:

1. $8x^4y^3$
2. $81n^4$
3. $4n^4$
4. $\frac{y^2}{x^2}$
5. $\frac{x^3}{4}$
6. $-\frac{1}{2}r^2$
7. $2b$