To solve the expression [tex]\(\left(7 m^{-2} m^2 n\right)^2\)[/tex] using the properties of exponents and the given values for [tex]\(m\)[/tex] and [tex]\(n\)[/tex], follow these steps:
1. Simplify the expression using exponent rules:
[tex]\[\left(7 m^{-2} m^2 n\right)^2\][/tex]
Notice that [tex]\(m^{-2} \cdot m^2\)[/tex] involves multiplying powers of the same base, [tex]\(m\)[/tex]:
[tex]\[m^{-2} \cdot m^2 = m^{(-2+2)} = m^0\][/tex]
Since any number raised to the power of zero is 1:
[tex]\[m^0 = 1\][/tex]
Therefore, the expression simplifies to:
[tex]\[\left(7 \cdot 1 \cdot n\right)^2 = (7n)^2\][/tex]
2. Substitute the given values of [tex]\(m\)[/tex] and [tex]\(n\)[/tex]:
Given [tex]\(m = -6\)[/tex] and [tex]\(n = 2\)[/tex], directly substitute [tex]\(n\)[/tex] in the simplified expression:
[tex]\[(7 \cdot 2)^2\][/tex]
3. Calculate the final value:
First, multiply within the parentheses:
[tex]\[7 \cdot 2 = 14\][/tex]
Then, square the result:
[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].
