Evaluate one sixth times the square root of 8.64 divided by 1.49.

Step-by-step solution

1. 1

   Read the commas as decimal points. In continental notation $8{,}64$ and $1{,}49$ mean $8.64$ and $1.49$ — they are not thousands separators and not a list of two numbers. So the expression is

   $\frac{1}{6}\sqrt{\frac{8.64}{1.49}}$

2. 2

   Simplify the fraction inside the radical. Clear the decimals by multiplying top and bottom by $100$:

   $\frac{8.64}{1.49}=\frac{864}{149}\approx 5.798658$

   Keeping $864/149$ as an exact fraction is what makes an exact answer possible ($149$ is prime, so this cannot reduce).

3. 3

   Split the radical and pull out the square factor. Since $864=144\cdot 6$ and $\sqrt{144}=12$,

   $\sqrt{\frac{864}{149}}=\frac{\sqrt{864}}{\sqrt{149}}=\frac{12\sqrt6}{\sqrt{149}}$

4. 4

   Divide by 6 and rationalise. The $12$ cancels against the $6$:

   $\frac{1}{6}\cdot\frac{12\sqrt6}{\sqrt{149}}=\frac{2\sqrt6}{\sqrt{149}}=\frac{2\sqrt{6\cdot 149}}{149}=\frac{2\sqrt{894}}{149}$

   That is the exact value, with no radical left in the denominator.

5. 5

   Get the decimal and check it backwards. $\sqrt{894}\approx 29.89983$, so

   $\frac{2(29.89983)}{149}\approx 0.401340$

   To verify, multiply by $6$ and square: $(6\cdot 0.401340)^2=2.40804^2\approx 5.79866$, which is exactly $8.64/1.49$. So the answer rounds to $0.4013$.