Factor out the least power of the variable or variable expression. Assume all variables represent positive real numbers. See Example 8. 2(3x+1)^-3/2+4(3x+1)^-1/2+6(3x+1)^1/2

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Identify the variable expression and its powers in each term. Here, the variable expression is $(3x+1)$ with powers $-\frac{3}{2}$, $-\frac{1}{2}$, and $\frac{1}{2}$ respectively.

Determine the least power (smallest exponent) among the terms. The powers are $-\frac{3}{2}$, $-\frac{1}{2}$, and $\frac{1}{2}$, so the least power is $-\frac{3}{2}$.

Factor out $(3x+1)^{-\frac{3}{2}}$ from each term. This means rewriting each term as a product of $(3x+1)^{-\frac{3}{2}}$ and another power of $(3x+1)$.

Express each term inside the parentheses after factoring out $(3x+1)^{-\frac{3}{2}}$ by subtracting the factored exponent from the original exponent. For example, for the second term, the new exponent is $-\frac{1}{2} - (-\frac{3}{2})$.

Write the factored expression as $(3x+1)^{-\frac{3}{2}}$ multiplied by the sum of the simplified terms inside the parentheses.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 powers inside the parentheses, making it easier to combine or simplify further.

Properties of Exponents

Understanding exponent rules, such as a^m / a^n = a^(m-n), is essential for manipulating terms with exponents. These properties allow you to rewrite expressions with fractional and negative exponents correctly during factoring.

Assumption of Positive Variables

Assuming variables represent positive real numbers ensures that expressions with fractional or negative exponents are well-defined and real-valued. This assumption allows the use of exponent rules without considering absolute values or complex numbers.

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