Factor out the least power of the variable or variable expression. Assume all variables represent positive real numbers. See Example 8.
$7(5t+3{)}^{-\frac{5}{3}}+14(5t+3{)}^{-\frac{2}{3}}-21(5t+3{)}^{\frac{1}{3}}$

Verified step by step guidance

1. Identify the common variable expression in all terms, which is ${(}^5$, ${(}^5$, and ${(}^5$.
2. Determine the least power (smallest exponent) among the powers of $(5t+3)$. The exponents are $\frac{-5}{3}$, $\frac{-2}{3}$, and $\frac{1}{3}$. The least power is $\frac{-5}{3}$.
3. Factor out ${(}^5$ from each term by rewriting each term as a product of this factor and another power of $(5t+3)$.
4. For each term, subtract the exponent of the factored out term from the original exponent to find the new exponent inside the parentheses after factoring. For example, for the second term, subtract $\frac{-5}{3}$ from $\frac{-2}{3}$ to get the new exponent.
5. Rewrite the expression as the product of the factored out term and the sum of the remaining terms with their adjusted exponents, keeping the coefficients intact.

Key Concepts

Factoring Expressions with Variable Exponents

Factoring expressions with variable exponents involves identifying the smallest exponent among terms and factoring it out. This simplifies the expression by reducing the powers inside the parentheses, making it easier to combine or simplify further.

Properties of Exponents

Understanding exponent rules, such as how to add, subtract, and factor exponents, is essential. For example, factoring out a term with an exponent means subtracting that exponent from each term's exponent inside the expression.

Assumption of Positive Variables

Assuming variables represent positive real numbers allows the use of exponent rules without worrying about undefined expressions or complex values. This assumption ensures that expressions with fractional or negative exponents are valid and simplifies the factoring process.