To simplify the algebraic expression \(\left(m^6 n\right)^3\), follow these steps:
1. Understand the Expression:
The expression \(\left(m^6 n\right)^3\) involves raising a product to a power. According to the properties of exponents, we can distribute the exponent over the product inside the parentheses.
2. Apply the Power to Each Factor:
The expression can be broken down, applying the power of 3 to each factor inside the parentheses:
[tex]\[
\left(m^6 n\right)^3 = (m^6)^3 \cdot (n)^3
\][/tex]
3. Calculate the New Exponents:
- For \(m^6\):
[tex]\[
(m^6)^3 = m^{6 \cdot 3} = m^{18}
\][/tex]
- For \(n\):
[tex]\[
n^3 = n^3
\][/tex]
4. Combine the Results:
The simplified expression, therefore, is:
[tex]\[
m^{18} \cdot n^3
\][/tex]
So, the expression \(\left(m^6 n\right)^3\) simplifies to:
[tex]\[
m^{18} n^3.
\][/tex]
Hence, the correct answer is:
[tex]\[
m^{18} n^3.
\][/tex]
