Rewrite each expression with positive exponents. Simplify, if possible. (a) \( \left(\frac{x^{-8} y^{4}}{3}\right)^{-3} \) (b) \( \left(\frac{a^{-6}}{b^{-6}}\right)\left(\frac{b}{a}\right)^{7} \) (a)

Let's tackle part (a) first! To rewrite \( \left(\frac{x^{-8} y^{4}}{3}\right)^{-3} \) with positive exponents, we first apply the negative exponent rule, which states that \( a^{-n} = \frac{1}{a^n} \). Thus, we can rewrite the expression as: \[ \left(\frac{3}{x^{-8} y^{4}}\right)^{3} = \frac{3^{3}}{(x^{-8})^{3} (y^{4})^{3}} = \frac{27}{x^{-24} y^{12}} \] Now changing \( x^{-24} \) into a positive exponent: \[ \frac{27 y^{12}}{x^{24}} \] So, the simplified expression with positive exponents is \( \frac{27 y^{12}}{x^{24}} \). Now for part (b): We begin with \( \left(\frac{a^{-6}}{b^{-6}}\right)\left(\frac{b}{a}\right)^{7} \). First, we can rewrite \( \frac{a^{-6}}{b^{-6}} \) as \( a^{-6} b^{6} \) (by flipping the denominator) and then apply the rule for negative exponents: \[ a^{-6} b^{6} = \frac{b^{6}}{a^{6}} \] Now, for the second part \( \left(\frac{b}{a}\right)^{7} \): \[ \left(\frac{b}{a}\right)^{7} = \frac{b^{7}}{a^{7}} \] Now, we can combine both parts: \[ \frac{b^{6}}{a^{6}} \cdot \frac{b^{7}}{a^{7}} = \frac{b^{6} b^{7}}{a^{6} a^{7}} = \frac{b^{13}}{a^{13}} \] So, the final simplified expression is \( \frac{b^{13}}{a^{13}} \).