simplify $\frac{6a^{2}b^{-2}}{8a^{-3}b^{3}}$. assume $a\neq0$ and $b\neq0$.\n$\frac{3a^{5}b^{-5}}{4}$\n$\frac…

simplify $\frac{6a^{2}b^{-2}}{8a^{-3}b^{3}}$. assume $a\neq0$ and $b\neq0$.\n$\frac{3a^{5}b^{-5}}{4}$\n$\frac{3b^{5}}{4a^{5}}$\n$\frac{3a^{5}}{4b^{5}}$\n$\frac{3a}{4b}$

simplify $\frac{6a^{2}b^{-2}}{8a^{-3}b^{3}}$. assume $a\neq0$ and $b\neq0$.\n$\frac{3a^{5}b^{-5}}{4}$\n$\frac{3b^{5}}{4a^{5}}$\n$\frac{3a^{5}}{4b^{5}}$\n$\frac{3a}{4b}$

Answer

Explanation:

Step1: Separate the coefficients and variables

Separate the coefficient - part $\frac{6}{8}$ and the variable - part $\frac{a^{2}}{a^{- 3}}\cdot\frac{b^{-2}}{b^{3}}$. $$\frac{6a^{2}b^{-2}}{8a^{-3}b^{3}}=\frac{6}{8}\cdot\frac{a^{2}}{a^{-3}}\cdot\frac{b^{-2}}{b^{3}}$$

Step2: Simplify the coefficient

Simplify $\frac{6}{8}$ to $\frac{3}{4}$. $$\frac{6}{8}=\frac{3}{4}$$

Step3: Use the quotient - rule of exponents for $a$

The quotient - rule of exponents is $\frac{x^{m}}{x^{n}}=x^{m - n}$. For $\frac{a^{2}}{a^{-3}}$, we have $a^{2-(-3)}=a^{2 + 3}=a^{5}$. $$\frac{a^{2}}{a^{-3}}=a^{2-(-3)}=a^{5}$$

Step4: Use the quotient - rule of exponents for $b$

For $\frac{b^{-2}}{b^{3}}$, we have $b^{-2-3}=b^{-5}=\frac{1}{b^{5}}$. $$\frac{b^{-2}}{b^{3}}=b^{-2 - 3}=b^{-5}$$

Step5: Combine the results

Multiply the simplified coefficient and variable parts together: $\frac{3}{4}\cdot a^{5}\cdot b^{-5}=\frac{3a^{5}b^{-5}}{4}$.

Answer:

$\frac{3a^{5}b^{-5}}{4}$