To simplify the expression \(\left(5^3\right)^2\) and rewrite it in the form \(5^n\), we will follow a detailed, step-by-step approach.
1. Understand the Expression:
\(\left(5^3\right)^2\) means that we are raising \(5^3\) to the power of 2.
2. Apply the Power of a Power Rule:
The power of a power rule states that \((a^m)^n = a^{m \cdot n}\). Here, \(a = 5\), \(m = 3\), and \(n = 2\).
3. Multiply the Exponents:
Following the rule \((a^m)^n = a^{m \cdot n}\), we have:
[tex]\[
(5^3)^2 = 5^{3 \cdot 2}
\][/tex]
Calculate the multiplication of the exponents:
[tex]\[
3 \cdot 2 = 6
\][/tex]
4. Rewrite the Expression:
Replace the exponent with the calculated value:
[tex]\[
(5^3)^2 = 5^6
\][/tex]
So, the simplified form of the expression \(\left(5^3\right)^2\) rewritten in the form \(5^n\) is \(5^6\).
To express the value:
[tex]\[
5^6 = 15625
\][/tex]
Final answer:
[tex]\[
\left(5^3\right)^2 = 5^6 = 15625
\][/tex]
