1. Indices: a) Simplify: \( \left(2^{3} \times 2^{4}\right) \div 2^{\text {}} \) b) Express \( 81^{1 / 4} \) in its simplest form.

To simplify \( \left(2^{3} \times 2^{4}\right) \div 2^{1} \), first apply the product of powers rule: \( 2^{3} \times 2^{4} = 2^{(3+4)} = 2^{7} \). Now divide by \( 2^{1} \): \( 2^{7} \div 2^{1} = 2^{(7-1)} = 2^{6} \). So the answer is \( 2^{6} \) or \( 64 \). To express \( 81^{1 / 4} \) in its simplest form, first notice that \( 81 = 3^{4} \). Therefore, we rewrite \( 81^{1/4} \) as \( (3^{4})^{1/4} \). Utilize the power of a power rule: \( (3^{4})^{1/4} = 3^{(4 \times \frac{1}{4})} = 3^{1} \). Thus, \( 81^{1/4} = 3 \).