Sure, let's simplify the given expression step-by-step.
### Expression to Simplify:
[tex]\[ \left(\frac{m^{-1} m^5}{m^{-2}}\right)^{-3} \][/tex]
### Step 1: Simplify the inner expression [tex]\(\frac{m^{-1} m^5}{m^{-2}}\)[/tex]
First, combine the exponents in the numerator:
[tex]\[ m^{-1} \cdot m^5 = m^{-1 + 5} = m^4 \][/tex]
So, the inner expression reduces to:
[tex]\[ \frac{m^4}{m^{-2}} \][/tex]
### Step 2: Further Simplify the Fraction
Use the properties of exponents to simplify the fraction:
[tex]\[ \frac{m^4}{m^{-2}} = m^{4 - (-2)} = m^{4 + 2} = m^6 \][/tex]
### Step 3: Raise the Result to the Power of -3
Now take the simplified inner expression and raise it to the power of [tex]\(-3\)[/tex]:
[tex]\[ \left(m^6\right)^{-3} = m^{6 \cdot (-3)} = m^{-18} \][/tex]
### Step 4: Express with Positive Exponents if Possible
To express [tex]\(m^{-18}\)[/tex] with a positive exponent, we write:
[tex]\[ m^{-18} = \frac{1}{m^{18}} \][/tex]
### Final Simplified Expression
Therefore, the simplified expression is:
[tex]\[ \frac{1}{m^{18}} \][/tex]
Among the given choices, the correct answer is:
[tex]\[ \frac{1}{m^{18}} \][/tex]
Thus, the final simplified form of the given expression is:
[tex]\[ \frac{1}{m^{18}} \][/tex]
