Factor out the least power of the variable or variable expression. Assume all variables represent positive real numbers. See Example 8. $6y^3(4y - 1)^{-3/7} - 8y^2(4y - 1)^{4/7} + 16y(4y - 1)^{11/7}$

Verified step by step guidance

1. Identify the variable expressions and their powers in each term: the terms are \$6y^{3}(4y-1)^{-3/7}\(, \)-8y^{2}(4y-1)^{4/7}\(, and \)16y^{1}(4y-1)^{11/7}$.
2. Determine the least power of each variable expression across all terms: for \(y\), the powers are 3, 2, and 1, so the least power is \(y^{1}\); for \((4y-1)\), the powers are \(-\frac{3}{7}\), \(\frac{4}{7}\), and \(\frac{11}{7}\), so the least power is \((4y-1)^{-\frac{3}{7}}\).
3. Factor out the least powers \(y^{1}(4y-1)^{-\frac{3}{7}}\) from each term, rewriting each term as a product of this factor and the remaining powers.
4. Express each term inside the parentheses after factoring out as follows: for the first term, divide by the factored out expression; for the second and third terms, subtract the exponents of the factored out powers from their original exponents.
5. Write the final factored expression as \(y(4y-1)^{-\frac{3}{7}}\) multiplied by the sum of the simplified terms inside the parentheses.

Key Concepts

Factoring Out the Least Power

Factoring out the least power involves identifying the smallest exponent of each variable or expression across all terms and extracting it as a common factor. This simplifies the expression by reducing the powers inside the parentheses, making it easier to work with or further simplify.

Properties of Exponents

Understanding exponent rules is essential, especially when dealing with variables raised to fractional or negative powers. Key rules include subtracting exponents when factoring out common terms and handling negative exponents by rewriting them as reciprocals.

Handling Variable Expressions with Exponents

When variables are part of expressions raised to powers, such as (4y - 1)^(m/n), treat the entire expression as a single base. Factoring requires comparing the fractional exponents and extracting the smallest power, ensuring all terms remain valid and simplified.