Factor out the least power of the variable or variable expression. Assume all variables represent positive real numbers. See Example 8.
$(3r+1{)}^{-\frac{2}{3}}+(3r+1{)}^{\frac{1}{3}}+(3r+1{)}^{\frac{4}{3}}$

Verified step by step guidance

1. Identify the variable expression and its powers in each term: the expression is $(3r+1)$ raised to the powers $\frac{-}{2}$, $\frac{1}{3}$, and $\frac{4}{3}$ respectively.
2. Determine the least power (smallest exponent) among the terms: compare $\frac{-}{2}$, $\frac{1}{3}$, and $\frac{4}{3}$ to find the smallest exponent, which will be the factor to take out.
3. Factor out the variable expression raised to the least power from each term: write the expression as ${(}^3$ raised to the least power, multiplied by the sum of the remaining terms with their exponents adjusted by subtracting the least power.
4. Rewrite each term inside the parentheses by subtracting the least power exponent from the original exponent: for example, if the least power is $\frac{-}{2}$, then the exponents inside become $\frac{0}{1}$ (which is 1), $\frac{1}{3}-\frac{-}{2}=\frac{1}{3}+\frac{2}{3}$, and so on.
5. Express the final factored form as the product of the variable expression raised to the least power and the simplified sum inside the parentheses.

Key Concepts

Exponent Rules

Exponent rules govern how to manipulate expressions with powers, including adding, subtracting, and factoring exponents. Understanding how to handle negative and fractional exponents is essential for simplifying and factoring expressions like (3r+1)^-2/3.

Factoring Expressions with Variables

Factoring involves rewriting an expression as a product of simpler expressions. When factoring variable expressions with different exponents, you factor out the term with the smallest exponent to simplify the expression effectively.

Properties of Positive Real Numbers

Assuming variables represent positive real numbers allows the use of properties like taking roots and fractional powers without concern for undefined or complex values. This assumption ensures that expressions with fractional exponents are well-defined and simplifies the factoring process.