Square Root worksheet

1. Cube root: 4.12

1. We're looking for the cube root of 70: a number, call it r, where multiplying 3 copies of r together gives 70

   $r^3$$=70$

2. No perfect cube except 1 divides 70, so nothing can come out from under the root sign. ∛70 is already as simple as it gets

   ${70}^{\frac{1}{3}}$$=\text{∛70}$

3. Work out the cube root as a decimal

   ${70}^{\frac{1}{3}}$$=4.121285$

4. Check the answer: multiplying 3 copies of it together should give back the original number (give or take rounding)

   ${4.121285}^3\approx 70$

2. Square root: 8.94

1. We're looking for the square root of 80: a number, call it r, where multiplying 2 copies of r together gives 80

   $r^2$$=80$

2. Estimate first: 80 lies between the perfect squares 64 = 8² and 81 = 9², so its square root is between 8 and 9

   $64<80<81$

3. To get the exact form, find the largest perfect square that divides 80. It's 16 = 4², so 4 comes out in front of the root sign

   $\sqrt{80}$$=\sqrt{16}\times \sqrt{5}$$=\text{4√5}$

4. Work out the square root as a decimal

   $\sqrt{80}$$=8.944272$

5. Check the answer: multiplying 2 copies of it together should give back the original number (give or take rounding)

   ${8.944272}^2\approx 80$

3. Cube root: 3.91

1. We're looking for the cube root of 60: a number, call it r, where multiplying 3 copies of r together gives 60

   $r^3$$=60$

2. No perfect cube except 1 divides 60, so nothing can come out from under the root sign. ∛60 is already as simple as it gets

   ${60}^{\frac{1}{3}}$$=\text{∛60}$

3. Work out the cube root as a decimal

   ${60}^{\frac{1}{3}}$$=3.914868$

4. Check the answer: multiplying 3 copies of it together should give back the original number (give or take rounding)

   ${3.914868}^3\approx 60$

4. Square root: 7.75

1. We're looking for the square root of 60: a number, call it r, where multiplying 2 copies of r together gives 60

   $r^2$$=60$

2. Estimate first: 60 lies between the perfect squares 49 = 7² and 64 = 8², so its square root is between 7 and 8

   $49<60<64$

3. To get the exact form, find the largest perfect square that divides 60. It's 4 = 2², so 2 comes out in front of the root sign

   $\sqrt{60}$$=\sqrt{4}\times \sqrt{15}$$=\text{2√15}$

4. Work out the square root as a decimal

   $\sqrt{60}$$=7.745967$

5. Check the answer: multiplying 2 copies of it together should give back the original number (give or take rounding)

   ${7.745967}^2\approx 60$

5. Square root: 7.07

1. We're looking for the square root of 50: a number, call it r, where multiplying 2 copies of r together gives 50

   $r^2$$=50$

2. Estimate first: 50 lies between the perfect squares 49 = 7² and 64 = 8², so its square root is between 7 and 8

   $49<50<64$

3. To get the exact form, find the largest perfect square that divides 50. It's 25 = 5², so 5 comes out in front of the root sign

   $\sqrt{50}$$=\sqrt{25}\times \sqrt{2}$$=\text{5√2}$

4. Work out the square root as a decimal

   $\sqrt{50}$$=7.071068$

5. Check the answer: multiplying 2 copies of it together should give back the original number (give or take rounding)

   ${7.071068}^2\approx 50$

6. 4th root: 2.84

1. We're looking for the 4th root of 65: a number, call it r, where multiplying 4 copies of r together gives 65

   $r^4$$=65$

2. No perfect 4th power except 1 divides 65, so nothing can come out from under the root sign. ∜65 is already as simple as it gets

   ${65}^{\frac{1}{4}}$$=\text{∜65}$

3. Work out the 4th root as a decimal

   ${65}^{\frac{1}{4}}$$=2.839412$

4. Check the answer: multiplying 4 copies of it together should give back the original number (give or take rounding)

   ${2.839412}^4\approx 65$

7. Cube root: 4.4

1. We're looking for the cube root of 85: a number, call it r, where multiplying 3 copies of r together gives 85

   $r^3$$=85$

2. No perfect cube except 1 divides 85, so nothing can come out from under the root sign. ∛85 is already as simple as it gets

   ${85}^{\frac{1}{3}}$$=\text{∛85}$

3. Work out the cube root as a decimal

   ${85}^{\frac{1}{3}}$$=4.39683$

4. Check the answer: multiplying 3 copies of it together should give back the original number (give or take rounding)

   ${4.39683}^3\approx 85$

8. 4th root: 2.77

1. We're looking for the 4th root of 59: a number, call it r, where multiplying 4 copies of r together gives 59

   $r^4$$=59$

2. No perfect 4th power except 1 divides 59, so nothing can come out from under the root sign. ∜59 is already as simple as it gets

   ${59}^{\frac{1}{4}}$$=\text{∜59}$

3. Work out the 4th root as a decimal

   ${59}^{\frac{1}{4}}$$=2.771488$

4. Check the answer: multiplying 4 copies of it together should give back the original number (give or take rounding)

   ${2.771488}^4\approx 59$

9. 6th root: 2.03

1. We're looking for the 6th root of 70: a number, call it r, where multiplying 6 copies of r together gives 70

   $r^6$$=70$

2. No perfect 6th power except 1 divides 70, so nothing can come out from under the root sign. ⁶√70 is already as simple as it gets

   ${70}^{\frac{1}{6}}$$=\text{⁶√70}$

3. Work out the 6th root as a decimal

   ${70}^{\frac{1}{6}}$$=2.030095$

4. Check the answer: multiplying 6 copies of it together should give back the original number (give or take rounding)

   ${2.030095}^6\approx 70$

10. 5th root: 2.27

1. We're looking for the 5th root of 60: a number, call it r, where multiplying 5 copies of r together gives 60

   $r^5$$=60$

2. No perfect 5th power except 1 divides 60, so nothing can come out from under the root sign. ⁵√60 is already as simple as it gets

   ${60}^{\frac{1}{5}}$$=\text{⁵√60}$

3. Work out the 5th root as a decimal

   ${60}^{\frac{1}{5}}$$=2.267933$

4. Check the answer: multiplying 5 copies of it together should give back the original number (give or take rounding)

   ${2.267933}^5\approx 60$