Question

which expression is equivalent to \\(\left(\frac{125^2}{125^{\frac{4}{3}}}\
ight)\\)? \
10 \
\\(\frac{1}{10}\\) \
\\(\frac{1}{25}\\) \
25

Explanation:

Step1: Use exponent rule for division

When dividing exponents with the same base, we subtract the exponents: $a^m \div a^n = a^{m - n}$. Here, the base is 125, so $\frac{125^2}{125^{\frac{4}{3}}}=125^{2 - \frac{4}{3}}$.

Step2: Calculate the exponent

First, simplify $2 - \frac{4}{3}$. Convert 2 to $\frac{6}{3}$, so $\frac{6}{3}-\frac{4}{3}=\frac{2}{3}$. Thus, we have $125^{\frac{2}{3}}$.

Step3: Rewrite 125 as a power of 5

Since $125 = 5^3$, substitute that in: $(5^3)^{\frac{2}{3}}$.

Step4: Use the power - of - a - power rule

The power - of - a - power rule states that $(a^m)^n=a^{m\times n}$. So, $(5^3)^{\frac{2}{3}} = 5^{3\times\frac{2}{3}}$.

Step5: Simplify the exponent

$3\times\frac{2}{3}=2$, so we have $5^2$.

Step6: Calculate $5^2$

$5^2 = 25$.

Answer:

25 (The option with "25" as its text)