Question

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

Explanation:

Step1: Use exponent rule for division

When dividing exponents with the same base, we subtract the exponents: \( \frac{a^m}{a^n} = a^{m - n} \). Here, the base \( a = 125 \), \( m = 2 \), and \( n=\frac{4}{3} \). So we first simplify the fraction inside the parentheses: \( 125^{2-\frac{4}{3}} \).

Step2: Calculate the exponent

Calculate \( 2-\frac{4}{3}=\frac{6}{3}-\frac{4}{3}=\frac{2}{3} \). So now we have \( 125^{\frac{2}{3}} \).

Step3: Rewrite 125 as a cube

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

Step4: Use exponent rule for powers of powers

When raising a power to a power, we multiply the exponents: \( (a^m)^n=a^{m\times n} \). 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