# How do you find the square root of 270?

Apr 29, 2017

See the solution process below:

#### Explanation:

We can use this rule of radicals to rewrite this expression:

$\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$

$\sqrt{270} = \sqrt{9 \cdot 30} = \sqrt{9} \cdot \sqrt{30} = \pm 3 \sqrt{30}$

If necessary, the $\sqrt{30} = \pm 5.477$

And therefore:

$\pm 3 \sqrt{30} = \pm 3 \cdot \pm 5.477 = \pm 16.432$ rounded to the nearest thousandth.

Apr 30, 2017

A = $3 \cdot \sqrt{30}$

#### Explanation:

Find a sqrt which can be divided by the whole number:
= $\sqrt{30}$

Then find the quotient of the number and the divisor (sqrt):
= 270 / $\sqrt{30}$
= 3
= 3 * $\sqrt{30}$

Apr 30, 2017

$\sqrt{270} = 3 \sqrt{30} \approx \frac{15873}{966} \approx 16.431677$

#### Explanation:

First note that $270$, like any positive number, has two square roots. We denote the positive square root by $\sqrt{270}$ and the negative one by $- \sqrt{270}$. However, the expression "the square root" is often used to refer to the principal, positive square root.

Next note that if $a , b \ge 0$ then:

$\sqrt{a b} = \sqrt{a} \sqrt{b}$

[[ The same is not true if both $a < 0$ and $b < 0$ ]]

Also, if $a \ge 0$ then:

$\sqrt{{a}^{2}} = a$

So we find:

$\sqrt{270} = \sqrt{9 \cdot 30} = \sqrt{9} \sqrt{30} = 3 \sqrt{30}$

This is the simplest form of the principal square root.

$3 \sqrt{30}$, like $\sqrt{30}$ is an irrational number.

We can calculate approximations to $\sqrt{30}$ using its continued fraction. Note that $30 = 5 \cdot 6$ is of the form $n \left(n + 1\right)$. Hence its square root has a short regular repeating pattern:

sqrt(30) = [5;bar(2,10)] = 5+1/(2+1/(10+1/(2+1/(10+1/(2+1/(10+...))))))

[[ In general sqrt(n(n+1)) = [n;bar(2,2n)] ]]

We can get decent approximations for $\sqrt{30}$ by truncating just before one of the $10$'s as follows:

$\sqrt{30} \approx 5 + \frac{1}{2} = \frac{11}{2}$

$\sqrt{30} \approx 5 + \frac{1}{2 + \frac{1}{10 + \frac{1}{2}}} = \frac{241}{44}$

$\sqrt{30} \approx 5 + \frac{1}{2 + \frac{1}{10 + \frac{1}{2 + \frac{1}{10 + \frac{1}{2}}}}} = \frac{5291}{966}$

Let's stop there and use this to give us an approximation for $\sqrt{270}$...

$\sqrt{270} = 3 \sqrt{30} \approx 3 \cdot \frac{5291}{966} = \frac{15873}{966} \approx 16.431677$