Question

find \\(\left(3x^{2}y^{-5}\
ight)^{3}\\).
\\(\bigcirc\\) a) \\(27x^{6}y^{15}\\)
\\(\bigcirc\\) b) \\(\dfrac{27x^{6}}{y^{15}}\\)
\\(\bigcirc\\) c) \\(\dfrac{9x^{6}}{y^{15}}\\)
\\(\bigcirc\\) d) \\(27x^{5}y^{2}\\)

Explanation:

Step1: Apply power of a product rule

The power of a product rule states that \((ab)^n = a^n b^n\). So, \((3x^{2}y^{-5})^{3}=3^{3}\times(x^{2})^{3}\times(y^{-5})^{3}\).

Step2: Calculate each term

• For the coefficient: \(3^{3}=27\).
• For the \(x\)-term: Using the power of a power rule \((a^{m})^{n}=a^{mn}\), we have \((x^{2})^{3}=x^{2\times3}=x^{6}\).
• For the \(y\)-term: Using the power of a power rule, \((y^{-5})^{3}=y^{-5\times3}=y^{-15}\). And since \(y^{-n}=\frac{1}{y^{n}}\), \(y^{-15}=\frac{1}{y^{15}}\).

Step3: Combine the terms

Multiplying the terms together, we get \(27\times x^{6}\times\frac{1}{y^{15}}=\frac{27x^{6}}{y^{15}}\).

Answer:

B) \(\frac{27x^{6}}{y^{15}}\)