Factor out the least power of the variable or variable expression. Assume all variables represent positive real numbers. See Example 8. 6r^-2/3-5r^-5/3

Verified step by step guidance

1. Identify the terms in the expression: \(6r^{-\frac{2}{3}} - 5r^{-\frac{5}{3}}\).
2. Determine the powers of \(r\) in each term: the first term has \(r^{-\frac{2}{3}}\) and the second term has \(r^{-\frac{5}{3}}\).
3. Find the least power (smallest exponent) of \(r\) between \(-\frac{2}{3}\) and \(-\frac{5}{3}\). Since \(-\frac{5}{3} < -\frac{2}{3}\), the least power is \(r^{-\frac{5}{3}}\).
4. Factor out \(r^{-\frac{5}{3}}\) from each term: write each term as a product involving \(r^{-\frac{5}{3}}\). For the first term, express \(r^{-\frac{2}{3}}\) as \(r^{-\frac{5}{3}} \times r^{\left(-\frac{2}{3} + \frac{5}{3}\right)}\).
5. Rewrite the expression as \(r^{-\frac{5}{3}}\) times the simplified terms inside parentheses, and simplify the exponents inside the parentheses to complete the factoring.

Key Concepts

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the positive exponent. For example, r^(-2/3) means 1 divided by r^(2/3). Understanding this helps in rewriting expressions to factor or simplify them correctly.

Factoring Out the Least Power

Factoring out the least power involves 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 with exponents during factoring. These rules are essential for combining, dividing, or factoring expressions involving variables with fractional or negative exponents.