Question

simplify the expression: \\(\frac{x^{-2}}{5y^{4}} \cdot \frac{5}{x^{3}y^{-4}}\\)

Explanation:

Step 1: Multiply the numerators and denominators

Multiply the coefficients and the variables separately. For the coefficients: \(1\times5 = 5\) (the coefficient of the first fraction is 1, since \(\frac{x^{-2}}{5y^4}\) can be thought of as \(1\times\frac{x^{-2}}{5y^4}\)). For the \(x\)-terms: \(x^{-2}\times x^{3}=x^{-2 + 3}\) (using the rule \(a^m\times a^n=a^{m + n}\)). For the \(y\)-terms: \(y^{4}\times y^{-4}=y^{4+( - 4)}\) (using the same rule for exponents). So we have \(\frac{5\times x^{-2+3}}{5\times y^{4+( - 4)}}\).

Step 2: Simplify the exponents

Simplify the exponents of \(x\) and \(y\). For \(x\): \(-2 + 3=1\), so \(x^{-2 + 3}=x^{1}=x\). For \(y\): \(4+( - 4)=0\), and any non - zero number to the power of 0 is 1, so \(y^{4+( - 4)}=y^{0}=1\) (assuming \(y
eq0\)). Also, simplify the coefficient: \(\frac{5}{5}=1\). Now we have \(\frac{1\times x\times1}{1\times1\times1}\).

Step 3: Final simplification

After simplifying the coefficient, \(x\)-term, and \(y\)-term, we get \(x\).

Answer:

\(x\)