1. **State the problem:** Calculate the value of the expression $$\left(\sqrt{0.25 \times 10^4}\right)\left(4 \times 10^3\right)^2$$. 2. **Recall formulas and rules:** - The square root of a product is the product of the square roots: $$\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$$. - When raising a product to a power: $$(ab)^n = a^n b^n$$. - Powers of 10: $$10^m \times 10^n = 10^{m+n}$$. 3. **Calculate the square root part:** $$\sqrt{0.25 \times 10^4} = \sqrt{0.25} \times \sqrt{10^4} = 0.5 \times 10^2 = 0.5 \times 100 = 50$$. 4. **Calculate the squared term:** $$\left(4 \times 10^3\right)^2 = 4^2 \times \left(10^3\right)^2 = 16 \times 10^{6}$$. 5. **Multiply the two results:** $$50 \times 16 \times 10^{6} = (50 \times 16) \times 10^{6} = 800 \times 10^{6}$$. 6. **Express in scientific notation:** $$800 \times 10^{6} = 8 \times 10^{2} \times 10^{6} = 8 \times 10^{8}$$. **Final answer:** $$8 \times 10^{8}$$.