1. We are asked to evaluate each expression without using a calculator. 2. (a) Evaluate $(-3)^4$: $$(-3)^4 = (-3) \times (-3) \times (-3) \times (-3) = 9 \times 9 = 81$$ 3. (b) Evaluate $-3^4$: Here, exponentiation occurs before applying the negative sign, so: $$-3^4 = -(3^4) = -(81) = -81$$ 4. (c) Evaluate $3^{-4}$: A negative exponent means reciprocal: $$3^{-4} = \frac{1}{3^4} = \frac{1}{81}$$ 5. (d) Evaluate $5^{\frac{23}{21}}$: This is an exponent power with a fraction. Since 23 and 21 are close, $$5^{\frac{23}{21}} = 5^{1 + \frac{2}{21}} = 5^1 \times 5^{\frac{2}{21}} = 5 \times 5^{\frac{2}{21}}$$ Without a calculator, we leave it in this simplified form. 6. (e) Evaluate $\left(\frac{2}{3}\right)^{-2}$: Applying negative exponent and power: $$\left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^2 = \frac{9}{4}$$ 7. (f) Evaluate $16^{-\frac{3}{4}}$: Rewrite 16 as $2^4$: $$16^{-\frac{3}{4}} = (2^4)^{-\frac{3}{4}} = 2^{4 \times -\frac{3}{4}} = 2^{-3} = \frac{1}{2^3} = \frac{1}{8}$$ Final answers: (a) 81 (b) -81 (c) \frac{1}{81} (d) 5 \times 5^{\frac{2}{21}} (e) \frac{9}{4} (f) \frac{1}{8}