Full solution

Q. Simplify. Express your answer using positive exponents.

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\frac{4k^2}{(2k^5)(k)}

Identify Given Expression: Write down the given expression and identify the coefficients and the powers of $k$ . The given expression is $\frac{4k^2}{(2k^5)(k)}$ . We have the coefficient $4$ in the numerator and the coefficients $2$ and $1$ (implied for $k$ ) in the denominator. We also have the powers of $k$ : $k^2$ in the numerator and $k^5$ and $k$ in the denominator.
Simplify Coefficients: Simplify the coefficients by dividing the coefficient in the numerator by the coefficients in the denominator. $\newline$ $\frac{4}{2} = 2$ . $\newline$ So, $\frac{4k^2}{(2k^5)(k)}$ becomes $\frac{2k^2}{(k^5)(k)}$ .
Simplify Powers of k: Simplify the powers of k by using the laws of exponents. $\newline$ When dividing powers with the same base, we subtract the exponents. $\newline$ $k^2$ divided by $k^5$ equals $k^{(2-5)}$ , which simplifies to $k^{-3}$ . $\newline$ Multiplying $k^{-3}$ by $k$ (which is $k^1$ ) gives us $k^{(-3+1)}$ , which simplifies to $k^{-2}$ . $\newline$ However, we made a mistake here. We should have multiplied $k^{-3}$ by $k$ after simplifying the division, not before. This is a math error.