select all the algebraic expressions equivalent to $\frac{a^{3}}{b^{3}}$.\n$a^{3}b^{3}$\n$(ab^{-1})^{3}$\n$\f…

select all the algebraic expressions equivalent to $\frac{a^{3}}{b^{3}}$.\n$a^{3}b^{3}$\n$(ab^{-1})^{3}$\n$\frac{a^{8}b^{2}}{a^{5}b^{5}}$\n$(\frac{3a3b}{6})^{2}$\n$a^{3}b^{-3}$

select all the algebraic expressions equivalent to $\frac{a^{3}}{b^{3}}$.\n$a^{3}b^{3}$\n$(ab^{-1})^{3}$\n$\frac{a^{8}b^{2}}{a^{5}b^{5}}$\n$(\frac{3a3b}{6})^{2}$\n$a^{3}b^{-3}$

Answer

Explanation:

Step1: Analyze option 1

$a^{3}b^{3}\neq\frac{a^{3}}{b^{3}}$

Step2: Analyze option 2

Using power - of - a - product rule $(ab^{-1})^{3}=a^{3}(b^{-1})^{3}=a^{3}b^{-3}=\frac{a^{3}}{b^{3}}$

Step3: Analyze option 3

Simplify $\frac{a^{8}b^{2}}{a^{5}b^{5}}$ using quotient rule of exponents $\frac{a^{m}}{a^{n}}=a^{m - n}$ and $\frac{b^{m}}{b^{n}}=b^{m - n}$. We get $\frac{a^{8-5}}{b^{5 - 2}}=\frac{a^{3}}{b^{3}}$

Step4: Analyze option 4

Simplify $(\frac{3a\times3b}{6})^{2}=(\frac{9ab}{6})^{2}=(\frac{3ab}{2})^{2}=\frac{9a^{2}b^{2}}{4}\neq\frac{a^{3}}{b^{3}}$

Step5: Analyze option 5

$a^{3}b^{-3}=\frac{a^{3}}{b^{3}}$

Answer:

$(ab^{-1})^{3}$, $\frac{a^{8}b^{2}}{a^{5}b^{5}}$, $a^{3}b^{-3}$