Propriedade do quociente

Quotient Property of Square Roots

The Quotient Property of Square Roots states that the square root of a quotient is equal to the quotient of the square roots of the dividend and divisor.

Here's another way to explain it:

For example: $$\begin{align} \sqrt{\frac{23}{7}}&= \frac{\sqrt{23}}{\sqrt{7}} \\ \sqrt{\frac43}&= \frac{\sqrt{4}}{\sqrt{3}} = \frac{2}{\sqrt{3}}\\ \sqrt{\frac{9}{25}}&= \frac{\sqrt{9}}{\sqrt{25}} = \frac{3}{5} \end{align}$$

$($It is important to note that you cannot split up radicals with sums and differences in the same way. In general, $\sqrt{x\pm y} \neq \sqrt{x}\pm \sqrt{y}.)$

Example walk through $\\$

Take a look at the square root below:

The number in the square root seems difficult to simplify. Let's try applying the quotient rule. I like to think about the quotient rule as "distributing" the square root to each component of the fraction. You may want to go back to the Division Property of Radicals to refresh your memory!

The quotient property is very similar to the product property. In fact, if we see $$\frac{121}{16}$$ as $$121\times \frac{1}{16}$$ The two rules are the same!

Now, let's look at a word problem. You are walking the distance that is the hypotenuse of the right triangle pictured below at a speed (rate) of $\sqrt{3}$ feet per minute. How long does walking this distance take you, given the following equation: $$\text{Distance}=\text{rate}\times\text{time}$$

To find the time it will take you to walk this distance, you must manipulate the equation $\text{Distance}=\text{rate}\times\text{time}$ or $d=rt$ for the $t$ variable. We see,

$$\begin{align} d&=rt\\ \frac{d}{r}&=t \end{align}$$

Now, we just have to plug in the numbers for distance and rate. We see,

$$\begin{align} t&=\frac{d}{r}\\ &=\frac{\sqrt{48}}{\sqrt{3}}\\ &=\sqrt{\frac{48}{3}}\\ &=\sqrt{16}\\ &=4 \end{align}$$

Therefore, it will take you $4$ minutes to walk $\sqrt{48}$ feet.

Comparison: Quotient Property & Raising Fractions to a Power$\\$

Remember how we raise a fraction to a power? The quotient property is essentially the same as that. Recall that taking the square root of a number is the same as taking that number to a power of $\frac{1}{2}$. $$\sqrt{\frac{121}{16}} =\left(\frac{121}{16}\right) ^{\frac{1}{2}}$$

We can "distribute" the power to each component of the fraction: $$\left(\frac{121}{16}\right) ^{\frac{1}{2}}=\frac{121^{\frac{1}{2}}}{16^{\frac{1}{2}}}$$

Then, we write the power in terms of square root: $$\frac{121^{\frac{1}{2}}}{16^{\frac{1}{2}}} = \frac{\sqrt{121}}{\sqrt{16}}$$

We have shown why the quotient property of square root works with a property we had previously learned about. This might help you understand how things work a little better. If you find this confusing, stick to the other explanations of the quotient property!

(Video) Quotient Rule and Simplifying Square Roots

by LarryHCC

Here's a video by LarryHCC showing you the quotient property for rational square roots and solving some problems.

$\large{\text{Summary}}$

The quotient property for square roots states that the quotient of two square roots is the same as the square root of the quotient. $$\frac{\sqrt a}{\sqrt b}=\sqrt{\frac{a}{b}}$$

Let's look at simplifying $\frac{\sqrt{18}}{\sqrt{2}}$. We can rewrite this as the square root of the quotient $\frac{18}{2}$. $$\frac{\sqrt{18}}{\sqrt2}=\sqrt{\frac{18}2}=\sqrt{9}=3$$

We can also go the other direction to simplify. If we have $\sqrt{\frac{36}{25}}$, we can turn it into the quotient of two square roots. $$\sqrt{\frac{36}{25}}=\frac{\sqrt{36}}{\sqrt{25}}=\frac65$$