Question

For $a,b\ne 0,$ select the correct statement(s).

• A. $\frac{a^m}{a^n}=a^{m-n}$
• B. $\frac{a^m}{a^n}=a^{\frac{m}{n}}$
• C. $\frac{a^n}{b^n}={\left(\frac{a}{b}\right)}^n$
• D. $\frac{a^n}{b^n}=\frac{a}{b}$

Answer 1

Answer: (A), (C)

$\frac{a^n}{b^n}={\left(\frac{a}{b}\right)}^n$

$\bullet$ Let's take an example:

$\frac{2^5}{2^2}=\frac{2\times 2\times 2\times 2\times 2}{2\times 2}=2\times 2\times 2=2^3\Rightarrow \frac{2^5}{2^2}=2^{5-2}$

When two exponents having same base and different powers are divided, it results in base raised to the difference between the two powers, i.e., $\underset{–}{\frac{a^m}{a^n}=a^{m-n}}.$

$\bullet$ Let's take an another example:

$\frac{4^3}{2^3}=\frac{4\times 4\times 4}{2\times 2\times 2}=8\Rightarrow \frac{4^3}{2^3}=(\frac{4}{2}{)}^3$

When two exponents having different bases and same power are divided, it results in the ratio of bases raised to the common power, i.e $\underset{–}{\frac{a^n}{b^n}={(\frac{a}{b})}^n}$.