1. **State the problem:** We are given the equation $$(a \times 10^b) \times (5.3 \times 10^8) = 2.226 \times 10^4$$ where $a \times 10^b$ is in standard form. We need to find the values of $a$ and $b$. 2. **Recall the rules for multiplying numbers in standard form:** - Multiply the decimal parts: $a \times 5.3$ - Add the powers of 10: $10^b \times 10^8 = 10^{b+8}$ 3. **Rewrite the equation using these rules:** $$a \times 5.3 \times 10^{b+8} = 2.226 \times 10^4$$ 4. **Equate the powers of 10:** Since the equation is in standard form, the powers of 10 must be equal: $$b + 8 = 4$$ 5. **Solve for $b$:** $$b = 4 - 8 = -4$$ 6. **Equate the decimal parts:** $$a \times 5.3 = 2.226$$ 7. **Solve for $a$:** $$a = \frac{2.226}{5.3}$$ 8. **Simplify the fraction:** $$a = \frac{\cancel{2.226}}{\cancel{5.3}} = 0.42$$ (approximate to two decimal places) 9. **Check if $a$ is between 1 and 10:** Since $a = 0.42$ is not between 1 and 10, adjust $a$ and $b$ to keep the number in standard form. 10. **Adjust $a$ and $b$:** Multiply $a$ by 10 and subtract 1 from $b$: $$a = 0.42 \times 10 = 4.2$$ $$b = -4 - 1 = -5$$ 11. **Final answer:** $$a = 4.2, \quad b = -5$$ This satisfies the standard form condition where $1 \leq a < 10$ and $b$ is an integer.