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 terms in the expression: \(-4a^{-\frac{2}{5}} + 16a^{-\frac{7}{5}}\).
2. Determine the least power (smallest exponent) of the variable \(a\) between the two terms. Compare \(-\frac{2}{5}\) and \(-\frac{7}{5}\) to find the smaller exponent.
3. Factor out \(a\) raised to the least power found in the previous step from both terms. This means rewriting each term as a product of \(a\) to the least power and another power of \(a\).
4. Express the original expression as the product of the factored out term and a binomial. For example, write it as \(a^{\text{least power}}\) times a quantity inside parentheses.
5. Simplify the coefficients and the powers of \(a\) inside the parentheses by subtracting exponents according to the rule \(a^{m} \div a^{n} = a^{m-n}\).

Key Concepts

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the corresponding positive exponent. For example, a^-n equals 1 divided by a^n. Understanding this helps in rewriting expressions to factor or simplify terms involving negative powers.

Factoring Out the Least Power

Factoring out the least power means identifying the smallest exponent of the variable among all terms and factoring it out as a common factor. This simplifies the expression by reducing the powers inside the parentheses and is essential for simplifying expressions with fractional or negative exponents.

Properties of Exponents with Fractional Powers

Fractional exponents represent roots, such as a^(m/n) meaning the nth root of a raised to the mth power. Recognizing how to manipulate fractional exponents allows for correct factoring and simplification, especially when exponents are negative fractions.