Question

fully simplify using only positive exponents.\\(\frac{5x^{7}y^{7}}{25x^{7}y^{2}}\\)

Explanation:

Step1: Simplify the coefficient

Simplify the fraction of the coefficients \(\frac{5}{25}\). We know that \(\frac{5}{25}=\frac{1}{5}\) since 5 divided by 25 is \(\frac{1}{5}\).

Step2: Simplify the \(x\)-terms

For the \(x\)-terms, we use the rule of exponents \(\frac{x^{m}}{x^{n}} = x^{m - n}\). Here, \(m = 7\) and \(n=7\), so \(\frac{x^{7}}{x^{7}}=x^{7 - 7}=x^{0}\). And any non - zero number to the power of 0 is 1, so \(x^{0} = 1\) (assuming \(x
eq0\)).

Step3: Simplify the \(y\)-terms

For the \(y\)-terms, we use the rule of exponents \(\frac{y^{m}}{y^{n}}=y^{m - n}\). Here, \(m = 7\) and \(n = 2\), so \(\frac{y^{7}}{y^{2}}=y^{7-2}=y^{5}\).

Step4: Combine the results

Multiply the results from the coefficient, \(x\)-terms, and \(y\)-terms together. We have \(\frac{1}{5}\times1\times y^{5}=\frac{y^{5}}{5}\).

Answer:

\(\frac{y^{5}}{5}\)