Question

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

Accepted Answer

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.

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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Key Concepts

Factoring Expressions with Variable Exponents

Properties of Exponents

Assumption of Positive Variables