Question

which expression is equivalent to \\(\frac{5y^3}{(5y)^{-2}}\\)? \\(\circ\\ y^5\\) \\(\circ\\ y^6\\) \\(\circ\\ 125y^3\\) \\(\circ\\ 125y^5\\)

Explanation:

Step1: Recall negative exponent rule

A negative exponent in the denominator can be moved to the numerator with a positive exponent. So, \(\frac{5y^{3}}{(5y)^{-2}} = 5y^{3}\times(5y)^{2}\)

Step2: Expand \((5y)^{2}\)

Using the power of a product rule \((ab)^n=a^n b^n\), we have \((5y)^{2}=5^{2}y^{2} = 25y^{2}\)

Step3: Multiply the terms

Now, multiply \(5y^{3}\) and \(25y^{2}\). For the coefficients: \(5\times25 = 125\). For the variables: \(y^{3}\times y^{2}=y^{3 + 2}=y^{5}\) (using the rule \(a^{m}\times a^{n}=a^{m + n}\))
So the expression simplifies to \(125y^{5}\)

Answer:

125\(y^{5}\) (the option corresponding to \(125y^{5}\))