Question

multiply.
$5x^{2}\cdot 4y^{3}x^{6}\cdot 2y$
simplify your answer as much as possible.

Explanation:

Step1: Group coefficients and like variables

Group the numerical coefficients (5, 4, 2) together and the like variables (\(x^2\) with \(x^6\), \(y^3\) with \(y\)) together. So we have \((5\times4\times2)\times(x^2\times x^6)\times(y^3\times y)\).

Step2: Multiply the coefficients

Multiply the numerical coefficients: \(5\times4\times2 = 40\).

Step3: Apply the product rule for exponents (\(a^m \cdot a^n = a^{m + n}\)) to \(x\) terms

For the \(x\) terms, \(x^2\times x^6 = x^{2 + 6}=x^8\) (using the rule \(a^m\cdot a^n=a^{m + n}\)).

Step4: Apply the product rule for exponents to \(y\) terms

For the \(y\) terms, \(y^3\times y = y^{3+1}=y^4\) (since \(y = y^1\), so using \(a^m\cdot a^n=a^{m + n}\) with \(m = 3\) and \(n = 1\)).

Step5: Combine the results

Multiply the results from Step2, Step3, and Step4 together: \(40\times x^8\times y^4 = 40x^8y^4\).

Answer:

\(40x^8y^4\)