To simplify and evaluate the expression [tex]\(\left(7 m^{-2} m^2 n \right)^2\)[/tex] for [tex]\(m = -6\)[/tex] and [tex]\(n = 2\)[/tex], follow these steps:
1. Simplify the expression using properties of exponents:
[tex]\[ \left(7 m^{-2} m^2 n \right)^2 \][/tex]
2. Using the properties of exponents, combine [tex]\(m^{-2} \cdot m^2\)[/tex]:
[tex]\[ m^{-2} \cdot m^2 = m^{-2+2} = m^0 \][/tex]
And we know that any number raised to the power of 0 is 1:
[tex]\[ m^0 = 1 \][/tex]
3. The expression now simplifies to:
[tex]\[ \left(7 \cdot 1 \cdot n \right)^2 \][/tex]
Which is:
[tex]\[ \left(7n\right)^2 \][/tex]
4. Substitute the given values of [tex]\(m\)[/tex] and [tex]\(n\)[/tex] into the simplified expression:
When [tex]\(n = 2\)[/tex]:
[tex]\[ \left(7 \cdot 2\right)^2 = (14)^2 \][/tex]
5. Evaluate [tex]\((14)^2\)[/tex]:
[tex]\[ (14)^2 = 196 \][/tex]
Therefore, when [tex]\(m = -6\)[/tex] and [tex]\(n = 2\)[/tex], the value of the expression is:
[tex]\[ 196 \][/tex]
