Question

simplify the expression \\(\frac{2m^{-4}}{(2m^{-2})^3}\\)

Explanation:

Step1: Simplify the denominator using power rule

First, simplify \((2m^{-2})^3\). Using the power of a product rule \((ab)^n = a^n b^n\) and the power of a power rule \((a^m)^n=a^{mn}\), we have:
\((2m^{-2})^3 = 2^3\times(m^{-2})^3=8m^{-6}\)

Step2: Rewrite the fraction and use exponent rules

The original expression is \(\frac{2m^{-4}}{(2m^{-2})^3}=\frac{2m^{-4}}{8m^{-6}}\). Now, use the quotient rule for exponents \(\frac{a^m}{a^n}=a^{m - n}\) and the quotient of coefficients \(\frac{2}{8}=\frac{1}{4}\).
For the coefficients: \(\frac{2}{8}=\frac{1}{4}\)
For the variable part: \(\frac{m^{-4}}{m^{-6}}=m^{-4-(-6)} = m^{2}\)

Step3: Combine the results

Multiply the coefficient and the variable part: \(\frac{1}{4}\times m^{2}=\frac{m^{2}}{4}\)

Answer:

\(\frac{m^{2}}{4}\)