1. Simplify fully, WITHOUT using a calculator. Your answers should have positive exponents a. \( \left(\frac{1}{2}\right)^{-3} \) b. \( 3^{-1} \cdot 2^{0} \) c. \( \left(\frac{\sqrt{x}}{\frac{-3}{x^{2}}}\right)^{\frac{1}{2}} \) d. \( \left(3 \frac{3}{8}\right)^{\frac{-2}{3}} \)

Sure, let's simplify each of these expressions step-by-step! a. \( \left(\frac{1}{2}\right)^{-3} \) To simplify this, we use the rule \( a^{-n} = \frac{1}{a^n} \). So, we have: \[ \left(\frac{1}{2}\right)^{-3} = \frac{1}{\left(\frac{1}{2}\right)^{3}} = \frac{1}{\frac{1}{8}} = 8. \] b. \( 3^{-1} \cdot 2^{0} \) Here, \( 2^{0} = 1 \) (any non-zero number to the power of 0 is 1). Then: \[ 3^{-1} \cdot 1 = \frac{1}{3}. \] c. \( \left(\frac{\sqrt{x}}{\frac{-3}{x^{2}}}\right)^{\frac{1}{2}} \) First, simplify the fraction: \[ \frac{\sqrt{x} \cdot x^2}{-3} = \frac{x^{2 + \frac{1}{2}}}{-3} = \frac{x^{\frac{5}{2}}}{-3}. \] Now apply the exponent: \[ \left(\frac{x^{\frac{5}{2}}}{-3}\right)^{\frac{1}{2}} = \frac{x^{\frac{5}{4}}}{\sqrt{-3}}. \] Since we want positive exponents, the answer is \( \frac{x^{\frac{5}{4}}}{\sqrt{3}i} \) (if dealing with complex numbers). d. \( \left(3 \frac{3}{8}\right)^{\frac{-2}{3}} \) First, convert the mixed number into an improper fraction: \[ 3 \frac{3}{8} = \frac{24 + 3}{8} = \frac{27}{8}. \] Now apply the negative exponent: \[ \left(\frac{27}{8}\right)^{\frac{-2}{3}} = \left(\frac{8}{27}\right)^{\frac{2}{3}}. \] Now take the cube root and square: \[ \frac{\sqrt[3]{8}}{\sqrt[3]{27}}^2 = \frac{2}{3}^2 = \frac{4}{9}. \] So, the simplified results are: a. 8 b. \( \frac{1}{3} \) c. \( \frac{x^{\frac{5}{4}}}{\sqrt{-3}} \) or \( \frac{x^{\frac{5}{4}}}{\sqrt{3} i} \) d. \( \frac{4}{9} \)