Factor out the least power of the variable or variable expression. Assume all variables represent positive real numbers. See Example 8. -4a^-2/5+16a^-7/5

Verified step by step guidance

1. Identify the common factor in the expression: \(-4a^{-\frac{2}{5}} + 16a^{-\frac{7}{5}}\).
2. The common factor is the term with the smallest exponent of \(a\), which is \(a^{-\frac{7}{5}}\).
3. Factor out \(a^{-\frac{7}{5}}\) from each term in the expression.
4. Rewrite each term by dividing by the common factor: \(-4a^{-\frac{2}{5}} = -4a^{-\frac{2}{5} + \frac{7}{5}}a^{-\frac{7}{5}}\) and \(16a^{-\frac{7}{5}} = 16a^{0}a^{-\frac{7}{5}}\).
5. Simplify the expression inside the parentheses after factoring out \(a^{-\frac{7}{5}}\).

Key Concepts

Factoring

Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. In algebra, this often involves identifying common terms or expressions that can be extracted from a polynomial or rational expression, simplifying the overall equation.

Exponents and Powers

Exponents represent repeated multiplication of a base number. Understanding how to manipulate exponents, including the rules for multiplying and dividing powers, is crucial in algebra. In this context, recognizing the least power of a variable allows for effective factoring and simplification of expressions.

Positive Real Numbers

Positive real numbers are all the numbers greater than zero, including integers, fractions, and irrational numbers. In algebra, assuming variables represent positive real numbers simplifies the analysis and manipulation of expressions, as it eliminates concerns about undefined operations, such as taking even roots of negative numbers.