Question

quiz 7.2.1 - rational exponent properties
simplify
$x^{-3}x^{2}y^{\frac{-1}{2}}$
\\(\circ\\) a. \\(\dfrac{1}{xy^{\frac{1}{2}}}\\)
\\(\circ\\) b. \\(\dfrac{x^{5}}{y^{\frac{1}{2}}}\\)
\\(\circ\\) c. \\(xy^{\frac{1}{2}}\\)
\\(\circ\\) d. \\(\dfrac{1}{x^{6}y^{\frac{1}{2}}}\\)

Explanation:

Step1: Combine like x terms

When multiplying terms with the same base, add exponents: $x^{-3} \times x^{2} = x^{-3+2} = x^{-1}$

Step2: Rewrite negative exponent

A negative exponent means reciprocal: $x^{-1} = \frac{1}{x}$

Step3: Combine with y term

Keep the $y^{-\frac{1}{2}}$ term, rewrite it as $\frac{1}{y^{\frac{1}{2}}}$, then multiply: $\frac{1}{x} \times \frac{1}{y^{\frac{1}{2}}} = \frac{1}{xy^{\frac{1}{2}}}$

Answer:

a. $\frac{1}{xy^{\frac{1}{2}}}$