Solving Exponential Expressions
To solve the given expression, we will use the properties of exponents:
The expression is \( \left( \frac{(5^3)^2}{(5^2)^4} \right) \).
Now, let's apply the power of a power property for exponents, which states that \((a^m)^n = a^{(m \cdot n)}\):
For the numerator, \((5^3)^2 = 5^{(3 \cdot 2)} = 5^6\),
For the denominator, \((5^2)^4 = 5^{(2 \cdot 4)} = 5^8\),
Now, rewrite the expression with the simplified exponents:
\( \frac{5^6}{5^8} \).
Using the quotient of powers property, which states that \(a^m/a^n = a^{(m-n)}\), we subtract the exponents:
\( 5^{(6-8)} = 5^{-2} \).
The negative exponent rule states that \(a^{-n} = \frac{1}{a^n}\):
Therefore, \(5^{-2} = \frac{1}{5^2}\).
Since \(5^2 = 25\), the expression simplifies to:
\( \frac{1}{25} \).
Therefore, the answer is option (1) \( \frac{1}{25} \).
