To simplify the given expression \(\left(8^3\right)^7\), we can use the property of exponents known as the power of a power. This property states that \((a^m)^n = a^{m \cdot n}\), where \(a\) is a base, and \(m\) and \(n\) are exponents.
1. Let's identify the base and the exponents in the given expression \(\left(8^3\right)^7\).
- The base \(a\) is 8.
- The first exponent \(m\) is 3.
- The second exponent \(n\) is 7.
2. Applying the power of a power property:
[tex]\[
\left(8^3\right)^7 = 8^{3 \cdot 7}
\][/tex]
3. Now, we perform the multiplication of the exponents:
[tex]\[
3 \cdot 7 = 21
\][/tex]
4. Therefore, the expression simplifies to:
[tex]\[
\left(8^3\right)^7 = 8^{21}
\][/tex]
In this case, \(n = 21\).
So, \(8^n = 8^{21}\), and the value of \(n\) is:
[tex]\[
n = 21
\][/tex]
