Question

simplify the following expression using the properties of exponents. the final form of the expressions with variables should contain only positive exponents. assume that all variables represent nonzero numbers.\\(\frac{25y^9 \cdot 3y^{-6}}{15xy^3}\\)

Explanation:

Step1: Multiply the coefficients and use exponent rule for multiplication in numerator

First, multiply the coefficients \(25\) and \(3\) in the numerator, and for the variable \(y\), use the property \(a^m\cdot a^n=a^{m + n}\). So we have:
\(\frac{25\times3\times y^{9+( - 6)}}{15xy^{3}}=\frac{75y^{3}}{15xy^{3}}\)

Step2: Simplify the coefficient and use exponent rule for division

Simplify the coefficient \(\frac{75}{15}=5\). For the variable \(y\), use the property \(\frac{a^m}{a^n}=a^{m - n}\). So we get:
\(\frac{5y^{3}}{xy^{3}} = 5\times\frac{y^{3}}{y^{3}}\times\frac{1}{x}\)

Step3: Simplify the \(y\) terms

Since \(\frac{y^{3}}{y^{3}}=y^{3 - 3}=y^{0} = 1\) (for \(y
eq0\)), the expression becomes:
\(5\times1\times\frac{1}{x}=\frac{5}{x}\)

Answer:

\(\frac{5}{x}\)