Question

perform the operation and simplify.\\(\frac{16x^3}{5y^9} cdot \frac{x^3y^7}{80xy^2}\\)\\(\frac{x^{?}}{square y^{square}}\\)

Explanation:

Step1: Multiply numerators and denominators

Multiply the numerators \(16x^{3}\) and \(x^{3}y^{7}\) to get \(16x^{3}\cdot x^{3}y^{7}=16x^{3 + 3}y^{7}=16x^{6}y^{7}\). Multiply the denominators \(5y^{9}\) and \(80xy^{2}\) to get \(5y^{9}\cdot80xy^{2}=400xy^{9+2}=400xy^{11}\). So the expression becomes \(\frac{16x^{6}y^{7}}{400xy^{11}}\).

Step2: Simplify coefficients and variables

Simplify the coefficient \(\frac{16}{400}=\frac{1}{25}\). For the \(x\) terms, use the rule \(a^{m}/a^{n}=a^{m - n}\), so \(x^{6}/x=x^{6 - 1}=x^{5}\). For the \(y\) terms, \(y^{7}/y^{11}=y^{7 - 11}=y^{-4}=\frac{1}{y^{4}}\). Combining these, we have \(\frac{1}{25}\cdot x^{5}\cdot\frac{1}{y^{4}}=\frac{x^{5}}{25y^{4}}\).

Answer:

The simplified form is \(\frac{x^{5}}{25y^{4}}\), so the exponent of \(x\) is \(5\), the coefficient in the denominator (the first box) is \(25\), and the exponent of \(y\) (the second box) is \(4\).