Factor out the least power of the variable or variable expression. Assume all variables represent positive real numbers. See Example 8. 9z^-1/2+2z^1/2

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Identify the terms in the expression: $9z^{-\frac{1}{2}} + 2z^{\frac{1}{2}}$.

Determine the powers of $z$ in each term: the first term has $z^{-\frac{1}{2}}$ and the second term has $z^{\frac{1}{2}}$.

Find the least power of $z$ between $-\frac{1}{2}$ and $\frac{1}{2}$, which is $-\frac{1}{2}$.

Factor out $z^{-\frac{1}{2}}$ from each term: write each term as a product involving $z^{-\frac{1}{2}}$.

Rewrite the expression as $z^{-\frac{1}{2}}$ times the sum of the remaining factors, simplifying the exponents inside the parentheses.

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Key Concepts

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

Exponent Rules

Exponent rules govern how to manipulate powers of variables, including multiplying, dividing, and factoring expressions with exponents. Understanding negative and fractional exponents is essential, as they represent roots and reciprocals, such as z^(-1/2) meaning 1 over the square root of z.

Factoring Expressions

Factoring involves rewriting an expression as a product of simpler expressions. When factoring out the least power of a variable, you identify the smallest exponent present and factor it out from each term, simplifying the expression and making it easier to work with.

Properties of Positive Real Numbers

Assuming variables represent positive real numbers allows the use of exponent rules without concern for undefined expressions, such as even roots of negative numbers. This assumption ensures that expressions like z^(1/2) are real and that factoring with fractional exponents is valid.

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