To simplify the expression [tex]\(\left(m^6 n\right)^3\)[/tex], we need to apply the properties of exponents systematically.
1. Identify the components inside the parentheses:
[tex]\(\left(m^6 n\right)\)[/tex].
2. Apply the exponent to each term inside the parentheses:
The expression inside the parentheses is [tex]\(m^6\)[/tex] and [tex]\(n\)[/tex].
3. Use the power of a product property [tex]\((ab)^c = a^c \cdot b^c\)[/tex]:
[tex]\(\left(m^6 n\right)^3 = (m^6)^3 \cdot (n)^3\)[/tex].
4. Simplify each part separately using the power of a power property [tex]\((a^b)^c = a^{b \cdot c}\)[/tex]:
[tex]\[
(m^6)^3 = m^{6 \cdot 3} = m^{18}
\][/tex]
[tex]\[
(n)^3 = n^3
\][/tex]
5. Combine the simplified parts:
[tex]\[
m^{18} \cdot n^3
\][/tex]
So, the simplified expression for [tex]\(\left(m^6 n\right)^3\)[/tex] is:
[tex]\[
m^{18} n^3
\][/tex]
Hence, the correct answer is [tex]\(m^{18} n^3\)[/tex].
