Question

(\frac{(2x^{2})^{3}x^{-4}}{x^{-3}y^{6}})

Explanation:

Step1: Simplify the numerator's power

Using the power of a product rule \((ab)^n = a^n b^n\) and power of a power rule \((a^m)^n = a^{mn}\), we have \((2x^2)^3 = 2^3\times(x^2)^3 = 8x^{6}\). So the numerator becomes \(8x^{6} \cdot x^{-4}\).

Step2: Combine like terms in numerator (x terms)

Using the rule \(a^m \cdot a^n = a^{m + n}\), for the x - terms: \(x^{6}\cdot x^{-4}=x^{6+( - 4)}=x^{2}\). So the numerator is now \(8x^{2}\).

Step3: Simplify the denominator

The denominator is \(x^{-3}\cdot y^{6}\).

Step4: Divide the numerator by the denominator

Using the rule \(\frac{a^m}{a^n}=a^{m - n}\) for x - terms and \(\frac{a^m}{a^n}=a^{m - n}\) for y - terms (we can consider the y - term in the numerator as \(y^{0}\) since there is no y in the numerator initially).
For x - terms: \(\frac{8x^{2}}{x^{-3}}=8x^{2-( - 3)} = 8x^{5}\)
For y - terms: \(\frac{y^{0}}{y^{6}}=y^{0 - 6}=y^{-6}=\frac{1}{y^{6}}\)
Combining these, we get \(\frac{8x^{5}}{y^{6}}\)

Answer:

\(\frac{8x^{5}}{y^{6}}\)