Multiplying and Dividing Square Roots

Look for the largest square number that divides 50. That is 25, because $50=25\times 2$.

Using the multiplication rule $\sqrt{a\times b}=\sqrt{a}\times \sqrt{b}$:

$\sqrt{50}=\sqrt{25}\times \sqrt{2}=5\sqrt{2}$.

The 25 must itself be square-rooted to give 5, and that 5 stays outside the root while the 2 stays inside, so the answer is $5\sqrt{2}\approx 7.07$. Both parts must be kept: the $\sqrt{2}$ cannot be dropped, and the 5 cannot be pushed back inside the root.

Look for the largest square number that divides 72. That is 36, because $72=36\times 2$.

Using $\sqrt{a\times b}=\sqrt{a}\times \sqrt{b}$:

$\sqrt{72}=\sqrt{36}\times \sqrt{2}=6\sqrt{2}$.

So the answer is $6\sqrt{2}\approx 8.49$. Check it: $6\sqrt{2}$ squared is $36\times 2=72$. Note that a square root is not half the number, the 6 must stay outside the root, and the $\sqrt{2}$ must not be dropped.

Use the division rule for square roots: $\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$.

So $\frac{\sqrt{50}}{\sqrt{2}}=\sqrt{\frac{50}{2}}=\sqrt{25}=5$.

Two things matter here: divide the right way round (50 by 2, not 2 by 50), and remember to take the square root at the end rather than stopping at 25.

You can check it the other way: $\sqrt{50}=5\sqrt{2}$, and $\frac{5\sqrt{2}}{\sqrt{2}}=5$.

Use the division rule for square roots: $\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$.

So $\frac{\sqrt{36}}{\sqrt{9}}=\sqrt{\frac{36}{9}}=\sqrt{4}=2$.

You can also work each root out separately: $\sqrt{36}=6$ and $\sqrt{9}=3$, and $6÷3=2$, the same answer.

Take care to divide 36 by 9 (not 9 by 36) and to take the square root of 4 at the end.

The root of a fraction applies to the top and the bottom: $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$.

So $\sqrt{\frac{16}{25}}=\frac{\sqrt{16}}{\sqrt{25}}=\frac{4}{5}$.

Both numbers must be rooted. Rooting only one of them leaves the fraction wrong. As a check, ${\left(\frac{4}{5}\right)}^2=\frac{16}{25}$, and $\frac{4}{5}$ is already in its simplest form.

Use the multiplication rule $\sqrt{a}\times \sqrt{b}=\sqrt{a\times b}$:

$\sqrt{0.63}\times \sqrt{100}=\sqrt{0.63\times 100}=\sqrt{63}$.

Take care with the decimal: multiplying by 100 moves the digits two places, so $0.63\times 100=63$, not 630.

Now simplify $\sqrt{63}$. The largest square factor of 63 is 9, since $63=9\times 7$, so $\sqrt{63}=\sqrt{9}\times \sqrt{7}=3\sqrt{7}\approx 7.94$. The $\sqrt{9}$ must be worked out as 3, and that 3 stays outside the root.