1. **Problem statement:** Estimate powers of 2 using the approximation $2^{10} \approx 1000$. 2. **Recall the rule:** $2^{a+b} = 2^a \times 2^b$ and $2^{ka} = (2^a)^k$. 3. **Calculate each:** **a) $2^{12}$** $$2^{12} = 2^{10} \times 2^2 \approx 1000 \times 4 = 4000$$ **b) $2^{25}$** (given) $$2^{25} = 2^5 \times 2^{10} \times 2^{10} = 32 \times 1000 \times 1000 = 32 \times 10^6 = 3.2 \times 10^7$$ **c) $2^{33}$** $$2^{33} = 2^{30} \times 2^3 = (2^{10})^3 \times 8 \approx 1000^3 \times 8 = 10^9 \times 8 = 8 \times 10^9$$ **d) $2^{104}$** $$2^{104} = 2^{100} \times 2^4 = (2^{10})^{10} \times 16 \approx 1000^{10} \times 16 = 10^{30} \times 16 = 1.6 \times 10^{31}$$ **e) $2^{200}$** $$2^{200} = (2^{10})^{20} \approx 1000^{20} = (10^3)^{20} = 10^{60}$$ **f) $2^{197}$** $$2^{197} = 2^{190} \times 2^7 = (2^{10})^{19} \times 128 \approx 1000^{19} \times 128 = 10^{57} \times 128 = 1.28 \times 10^{59}$$ --- 4. **Simplify expressions:** **a) $(2^{3a+1})^2$** $$= 2^{2(3a+1)} = 2^{6a+2}$$ **b) $\left(\frac{3^{1-2a}}{3^{a+2}}\right)^4$** $$= \left(3^{1-2a-(a+2)}\right)^4 = \left(3^{1-2a - a - 2}\right)^4 = \left(3^{-3a -1}\right)^4 = 3^{4(-3a -1)} = 3^{-12a -4}$$ **c) $5^{3^{2-a}}$** (assuming exponentiation is $5^{3^{2-a}}$) No simplification without more info. **d) $[7^{2(3-a)}]^3$** $$= 7^{3 \times 2(3-a)} = 7^{6(3-a)} = 7^{18 - 6a}$$ --- 5. **Solve for $x$:** **a) $\frac{2^{26}}{2^x} = 2^x$** $$2^{26 - x} = 2^x$$ Equate exponents: $$26 - x = x$$ $$26 = 2x$$ $$x = 13$$ **b) $\frac{2^x}{2^9} = 2^{2x}$** $$2^{x - 9} = 2^{2x}$$ Equate exponents: $$x - 9 = 2x$$ $$-9 = x$$ **c) $\frac{9^{2x}}{27^2} = 3^{2x}$** Rewrite bases as powers of 3: $$9 = 3^2, \quad 27 = 3^3$$ $$\frac{(3^2)^{2x}}{(3^3)^2} = 3^{2x}$$ $$\frac{3^{4x}}{3^6} = 3^{2x}$$ $$3^{4x - 6} = 3^{2x}$$ Equate exponents: $$4x - 6 = 2x$$ $$2x = 6$$ $$x = 3$$ **d) $\frac{4^{3x}}{8^x} = 2^7$** Rewrite bases as powers of 2: $$4 = 2^2, \quad 8 = 2^3$$ $$\frac{(2^2)^{3x}}{(2^3)^x} = 2^7$$ $$\frac{2^{6x}}{2^{3x}} = 2^7$$ $$2^{6x - 3x} = 2^7$$ $$2^{3x} = 2^7$$ Equate exponents: $$3x = 7$$ $$x = \frac{7}{3}$$