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

Verified step by step guidance

1. Identify the terms in the expression: $\frac{3}{m^{\frac{2}{3}}}-\frac{4}{m^{\frac{-}{1}}}$.
2. Determine the exponents of $m$ in each term: the first term has exponent $\frac{2}{3}$, and the second term has exponent $\frac{-}{1}$.
3. Find the least (smallest) exponent between $\frac{2}{3}$ and $\frac{-}{1}$, which is $\frac{-}{1}$.
4. Factor out $m^{\frac{-}{1}}$ from each term by rewriting each term as a product of $m^{\frac{-}{1}}$ and the remaining power of $m$.
5. Express the factored form as $m^{\frac{-}{1}}\left(\frac{3}{m^{\frac{2}{3}}}\right)-\frac{4}{1}$, simplifying the exponents inside the parentheses.

Key Concepts

Exponent Rules

Exponent rules govern how to manipulate powers of variables, including multiplication, division, and taking powers of powers. Understanding negative and fractional exponents is essential, as they represent reciprocals and roots respectively, which helps in simplifying expressions like m^(2/3) and m^(-1/3).

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, making it easier to work with or further simplify.

Assumption of Positive Variables

Assuming variables represent positive real numbers allows the use of fractional exponents without considering absolute values. This assumption simplifies the manipulation of roots and powers, ensuring expressions like m^(1/3) are well-defined and positive.