Approximation Problems

For these questions, you are asked to demonstrate that the square root of the given number matches the provided decimal approximation:

1. Show that $\sqrt{6} = 2.449$ (approx).

2. Show that $\sqrt{7} = 2.646$ (approx).

3. Show that $\sqrt{10} = 3.162$ (approx).

4. Show that $\sqrt{11} = 3.317$ (approx).

5. Show that $\sqrt{13} = 3.606$ (approx).

Simplification Problems

These problems require you to simplify the radical into the form $a\sqrt{b}$: 6. Simplify $\sqrt{20}$. 7. Simplify $\sqrt{45}$. 8. Simplify $\sqrt{72}$.

Calculation Problems

In these tasks, you must use a known square root value to find the value of a larger radical: 9. If $\sqrt{5} = 2.236$ (approx), find the value of $\sqrt{125}$ correct to two decimal places. 10. If $\sqrt{3} = 1.732$ (approx), find the value of $\sqrt{27}$ correct to two decimal places

Square Root Approximations

To “show” these values without a calculator, you can multiply the decimal by itself to see if it results in the number under the radical.

1. Show that $\sqrt{6} \approx 2.449$: Multiply $2.449 \times 2.449 = 5.997601$. Since this is nearly $6$, the approximation is correct.

2. Show that $\sqrt{7} \approx 2.646$: Multiply $2.646 \times 2.646 = 7.001316$. This confirms the value is approximately $7$.

3. Show that $\sqrt{10} \approx 3.162$: Multiply $3.162 \times 3.162 = 9.998244$, which rounds to $10$.

4. Show that $\sqrt{11} \approx 3.317$: Multiply $3.317 \times 3.317 = 11.002489$, confirming the approximation for $11$.

5. Show that $\sqrt{13} \approx 3.606$: Multiply $3.606 \times 3.606 = 13.003236$, which is approximately $13$.

Simplifying Radicals ($a\sqrt{b}$)

To simplify, find the largest perfect square (like $4, 9, 16, 25, 36$) that divides into the number.

6. Simplify $\sqrt{20}$:

   • Find factors: $20 = 4 \times 5$.

   • Apply the root: $\sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5}$.

   • Result: $2\sqrt{5}$.

7. Simplify $\sqrt{45}$:

   • Find factors: $45 = 9 \times 5$.

   • Apply the root: $\sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5}$.

   • Result: $3\sqrt{5}$.

8. Simplify $\sqrt{72}$:

   • Find factors: $72 = 36 \times 2$.

   • Apply the root: $\sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2}$.

   • Result: $6\sqrt{2}$.

Using Known Values

These problems require you to simplify the large radical first, then substitute the provided decimal.

9. If $\sqrt{5} \approx 2.236$, find $\sqrt{125}$:

   • Simplify $\sqrt{125}$ into $\sqrt{25 \times 5}$, which is $5\sqrt{5}$.

   • Substitute: $5 \times 2.236 = 11.18$.

   • Result: $11.18$.

10. If $\sqrt{3} \approx 1.732$, find $\sqrt{27}$:

    • Simplify $\sqrt{27}$ into $\sqrt{9 \times 3}$, which is $3\sqrt{3}$.

    • Substitute: $3 \times 1.732 = 5.196$.

    • Round to two decimal places: $5.20$.

    • Result: $5.20$.