Approximating Roots to One Decimal Place

Example 3: Approximating Square Root of 11

Steps:

1. Identify Perfect Squares:

   • Know perfect squares near 11.
   • (2^2 = 4), (3^2 = 9), (4^2 = 16).

2. Determine Bounds:

   • The square root of 11 is between (3^2) and (4^2) (i.e., 9 and 16).
   • Write: (\sqrt{9} < \sqrt{11} < \sqrt{16}).

3. Calculate Units Between:

   • Units between 9 and 11: 2 units.
   • Units between 11 and 16: 5 units.

4. Use Ratio to Approximate Decimal:

   • Calculate: (\frac{2}{2+5} = \frac{2}{7} \approx 0.3).

5. Approximate Square Root:

   • First digit: 3 (since it is between 3 and 4).
   • Decimal: 0.3 (from ratio).
   • Approximation: (\sqrt{11} \approx 3.3).

Example: Approximating Cube Root of 140

Steps:

1. Identify Perfect Cubes:

   • Know perfect cubes near 140.
   • (2^3 = 8), (3^3 = 27), (4^3 = 64), (5^3 = 125), (6^3 = 216).

2. Determine Bounds:

   • Cube root of 140 is between (5^3) and (6^3) (i.e., 125 and 216).
   • Write: (\sqrt[3]{125} < \sqrt[3]{140} < \sqrt[3]{216}).

3. Calculate Units Between:

   • Units between 125 and 140: 15 units.
   • Units between 140 and 216: 76 units.

4. Use Ratio to Approximate Decimal:

   • Calculate: (\frac{15}{15+76} = \frac{15}{91} \approx 0.2).

5. Approximate Cube Root:

   • First digit: 5 (since it is between 5 and 6).
   • Decimal: 0.2 (from ratio).
   • Approximation: (\sqrt[3]{140} \approx 5.2).