The document shows steps to rearrange an equation to solve for ν. It takes the nth root of both sides, raises both sides to a power, multiplies both sides by a term, and takes the cube root of both sides to isolate ν. It then replaces the terms in the original equation with their units to verify that the unit of the Kolmogorov length scale is meters.

1. 2a Rearrange for ν.
휂 =
휈3
휖
1
4

2. 2a • Remember that taking the nth root of something is the same as raising it to the power 1/n:

3. 2a • Remember that taking the nth root of something is the same as raising it to the power 1/n:
η =
4 ν3
ϵ

4. 2a • Remember that taking the nth root of something is the same as raising it to the power 1/n:
η =
4 ν3
ϵ
• Raise both sides to the power 4:

5. 2a • Remember that taking the nth root of something is the same as raising it to the power 1/n:
η =
4 ν3
ϵ
• Raise both sides to the power 4:
η4 =
ν3
ϵ

6. 2a • Remember that taking the nth root of something is the same as raising it to the power 1/n:
η =
4 ν3
ϵ
• Raise both sides to the power 4:
η4 =
ν3
ϵ
• Multiply both sides by ϵ:

7. 2a • Remember that taking the nth root of something is the same as raising it to the power 1/n:
η =
4 ν3
ϵ
• Raise both sides to the power 4:
η4 =
ν3
ϵ
• Multiply both sides by ϵ:
ϵη4 = ν3

8. 2a • Remember that taking the nth root of something is the same as raising it to the power 1/n:
η =
4 ν3
ϵ
• Raise both sides to the power 4:
η4 =
ν3
ϵ
• Multiply both sides by ϵ:
ϵη4 = ν3
• Take the cube root of both sides:

9. 2a • Remember that taking the nth root of something is the same as raising it to the power 1/n:
η =
4 ν3
ϵ
• Raise both sides to the power 4:
η4 =
ν3
ϵ
• Multiply both sides by ϵ:
ϵη4 = ν3
• Take the cube root of both sides:
ퟑ 훜훈ퟒ = 훎

10. 2b By replacing the components of the original equation with
their units, verify that the unit of the Kolmogorov length
scale is metres.

11. 2b • Replace all elements of the original equation with their units:
η =
ν3
ϵ
1
4

12. 2b • Replace all elements of the original equation with their units:
η =
ν3
ϵ
1
4
η =
m2s−1 3
m2s−3
1
4

13. 2b • Replace all elements of the original equation with their units:
η =
ν3
ϵ
1
4
η =
m2s−1 3
m2s−3
1
4
• Multiply out the top of the fraction:

14. 2b • Replace all elements of the original equation with their units:
η =
ν3
ϵ
1
4
η =
m2s−1 3
m2s−3
1
4
• Multiply out the top of the fraction:
(η) =
4 m2m2m2s−1s−1s−1
m2s−3

15. 2b • Replace all elements of the original equation with their units:
η =
ν3
ϵ
1
4
η =
m2s−1 3
m2s−3
1
4
• Multiply out the top of the fraction:
(η) =
4 m2m2m2s−1s−1s−1
m2s−3
• Combine and cancel like units and simplify:

16. 2b • Replace all elements of the original equation with their units:
η =
ν3
ϵ
1
4
η =
m2s−1 3
m2s−3
1
4
• Multiply out the top of the fraction:
(η) =
4 m2m2m2s−1s−1s−1
m2s−3
• Combine and cancel like units and simplify:
η =
4 m6s−3
m2s−3 =
4
m4 = 퐦