<p>What is the digit in the unit's place of <span>\((1354)^{22222}\)</span>?</p>
Step-by-Step Solution
Key Concept: Unit digits of powers follow periodic patterns. Find the period, divide the exponent by the period, and use the remainder to determine the unit digit.
<p><strong>Step 1:</strong> Identify the last digit of the base: The last digit of 1354 is 4.</p><p><strong>Step 2:</strong> Determine the period of powers of 4: The unit digits of powers of 4 follow the pattern: 4, 6, 4, 6, ... (period = 2).</p><p><strong>Step 3:</strong> Find the remainder when the exponent is divided by the period: $22222 \div 2$ gives remainder 0.</p><p><strong>Step 4:</strong> Use the switch number: When remainder is 0, the switch number is 0, which corresponds to the unit digit 6.</p><p>∴ The digit in the unit's place of $(1354)^{22222}$ is <strong>6</strong>.</p>
Correct Answer: 6