Question: 3. Evaluate.
(a) $(-3)^4$
(b) $4^2 \times 4$
(c) $\left(\frac{2^4}{4^2}\right)^2$
(d) $5^2 \div (5 \times 2^{-1})^3$
1. **State the problem:** Evaluate each expression.
2. **Part (a):** Evaluate $(-3)^4$.
- The expression means $(-3)$ multiplied by itself 4 times.
- Calculate: $$(-3)^4 = (-3) \times (-3) \times (-3) \times (-3)$$
- Multiply stepwise: $$(-3) \times (-3) = 9$$
- Then: $$9 \times (-3) = -27$$
- Finally: $$-27 \times (-3) = 81$$
3. **Part (b):** Evaluate $4^2 \times 4$.
- Calculate $4^2$ first: $$4^2 = 16$$
- Then multiply by 4: $$16 \times 4 = 64$$
4. **Part (c):** Evaluate $\left(\frac{2^4}{4^2}\right)^2$.
- Calculate numerator: $$2^4 = 16$$
- Calculate denominator: $$4^2 = 16$$
- Form the fraction: $$\frac{16}{16} = 1$$
- Raise to the power 2: $$1^2 = 1$$
5. **Part (d):** Evaluate $5^2 \div (5 \times 2^{-1})^3$.
- Calculate $5^2$: $$5^2 = 25$$
- Calculate $2^{-1}$: $$2^{-1} = \frac{1}{2}$$
- Multiply inside parentheses: $$5 \times \frac{1}{2} = \frac{5}{2}$$
- Raise to the power 3: $$\left(\frac{5}{2}\right)^3 = \frac{5^3}{2^3} = \frac{125}{8}$$
- Now divide: $$25 \div \frac{125}{8} = 25 \times \frac{8}{125}$$
- Simplify: $$25 = 5 \times 5, \quad 125 = 5 \times 5 \times 5$$
- Cancel common factors: $$25 \times \frac{8}{125} = \cancel{25} \times \frac{8}{\cancel{125}} = 1 \times \frac{8}{5} = \frac{8}{5}$$
**Final answers:**
(a) $81$
(b) $64$
(c) $1$
(d) $\frac{8}{5}$