Question

Choose the correct expression(s).

• A. $\frac{5^{2025}}{5^{2021}}=5^4$
• B. $\frac{5^{2021}}{{10}^{2021}}=(\frac{1}{2}{)}^{2021}$
• C. $\frac{{25}^{1011}}{6^{2022}}=(\frac{5}{6}{)}^{2022}$

Answer 1

Answer: (A), (B), (C)

$\frac{{25}^{1011}}{6^{2022}}=(\frac{5}{6}{)}^{2022}$

Let's check all the given expressions one-by-one:

$\bullet \frac{5^{2025}}{5^{2021}}$

$=5^{2025-2021}=5^4$
[ <!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> Division rule of exponents: $\frac{a^m}{a^n}=a^{m-n}$ ]

$\bullet \frac{5^{2021}}{{10}^{2021}}$

$=(\frac{5}{10}{)}^{2021}=(\frac{1}{2}{)}^{2021}$
[ <!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> Division rule of exponents: $\frac{a^n}{b^n}=(\frac{a}{b}{)}^n$ ]

$\bullet \frac{{25}^{1011}}{6^{2022}}$
Here, neither bases nor exponents are same. Division rule of exponents can not be applied directly.

Let's try to make exponents same.

$\frac{{25}^{1011}}{6^{2022}}$
$=\frac{(5^2{)}^{1011}}{6^{2022}}$
$(\because 25=5\times 5=5^2)$

$=\frac{5^{2\times 1011}}{6^{2022}}$
[ <!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> Repeated exponentiation: $(a^m{)}^n=a^{m\times n}$ ]

$=\frac{5^{2022}}{6^{2022}}$

$\Rightarrow \frac{{25}^{1011}}{6^{2022}}=(\frac{5}{6}{)}^{2022}$
[ <!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> Division rule of exponents: $\frac{a^n}{b^n}=(\frac{a}{b}{)}^n$ ]

Therefore, all the given expressions are correct.