📘 Quiz

Solution: ## Step 1: Analyze the given expressions\nThe expressions given are: \$\u221a75 - \u221a74\$, \$\u221a74 - \u221a73\$, \$\u221a76 - \u221a75\$, and \$\u221a77 - \u221a76\$.\n\n## Step 2: Rationalize each expression\nTo compare these, let's rationalize each expression by multiplying with its conjugate over itself: \$\frac{(\u221a75 - \u221a74)(\u221a75 + \u221a74)}{\u221a75 + \u221a74}\$, \$\frac{(\u221a74 - \u221a73)(\u221a74 + \u221a73)}{\u221a74 + \u221a73}\$, \$\frac{(\u221a76 - \u221a75)(\u221a76 + \u221a75)}{\u221a76 + \u221a75}\$, and \$\frac{(\u221a77 - \u221a76)(\u221a77 + \u221a76)}{\u221a77 + \u221a76}\$.\n\n## Step 3: Simplify the expressions\nAfter rationalization, the expressions become: \$\frac{75-74}{\u221a75 + \u221a74}\$, \$\frac{74-73}{\u221a74 + \u221a73}\$, \$\frac{76-75}{\u221a76 + \u221a75}\$, and \$\frac{77-76}{\u221a77 + \u221a76}\$.\n\n## Step 4: Further simplification\nThis simplifies to: \$\frac{1}{\u221a75 + \u221a74}\$, \$\frac{1}{\u221a74 + \u221a73}\$, \$\frac{1}{\u221a76 + \u221a75}\$, and \$\frac{1}{\u221a77 + \u221a76}\$.\n\n## Step 5: Compare the denominators\nThe larger the denominator, the smaller the fraction. So, we compare \u221a75 + \u221a74, \u221a74 + \u221a73, \u221a76 + \u221a75, and \u221a77 + \u221a76.\n\n## Step 6: Analyze the square root values\nApproximate values are: \u221a75 \u2248 8.66, \u221a74 \u2248 8.60, \u221a73 \u2248 8.54, \u221a76 \u2248 8.72, and \u221a77 \u2248 8.77.\n\n## Step 7: Calculate the sums\nThe sums are approximately: 8.66 + 8.60 = 17.26, 8.60 + 8.54 = 17.14, 8.72 + 8.66 = 17.38, and 8.77 + 8.72 = 17.49.\n\n## Step 8: Determine the largest denominator\nThe largest denominator is 17.49 (\u221a77 + \u221a76) and the smallest is 17.14 (\u221a74 + \u221a73).\n\n## Step 9: Identify the smallest expression\nThe smallest expression is \$\frac{1}{\u221a77 + \u221a76}\$, which corresponds to \$\u221a77 - \u221a76\$.\n\nThe final answer is: \$\u221a77 - \u221a76\$