Factor out the least power of the variable or variable expression. Assume all variables represent positive real numbers. See Example 8. 4k^-1+k^-2

Verified step by step guidance

1. Identify the terms in the expression: \$4k^{-1} + k^{-2}$.
2. Determine the powers of \(k\) in each term: the first term has \(k^{-1}\) and the second term has \(k^{-2}\).
3. Find the least power of \(k\) between \(-1\) and \(-2\), which is \(-2\).
4. Factor out \(k^{-2}\) from each term: write the expression as \(k^{-2}(4k^{(-1) - (-2)} + k^{(-2) - (-2)})\).
5. Simplify the exponents inside the parentheses: \(k^{-2}(4k^{1} + 1)\), which is \(k^{-2}(4k + 1)\).

Key Concepts

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the positive exponent. For example, k^-1 equals 1/k, and k^-2 equals 1/k^2. Understanding this helps in rewriting expressions to factor or simplify them.

Factoring Out the Least Power

Factoring out the least power means identifying the smallest exponent of the variable in all terms and factoring it out as a common factor. This simplifies the expression by reducing the powers inside the parentheses.

Properties of Exponents

Properties of exponents, such as a^m / a^n = a^(m-n), allow manipulation of terms when factoring. These rules help rewrite terms after factoring out the common variable power, making the expression easier to handle.