How do you find the square root of 270?

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How do you find the square root of 270?

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Answer 1 · 2017-04-29T23:59:03

See the solution process below:

Explanation:

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

$\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$ $sqrt(a * b) = sqrt(a) * sqrt(b)$

$\sqrt{270} = \sqrt{9 \cdot 30} = \sqrt{9} \cdot \sqrt{30} = \pm 3 \sqrt{30}$ $sqrt(270) = sqrt(9 * 30) = sqrt(9) * sqrt(30) = +-3sqrt(30)$

If necessary, the $\sqrt{30} = \pm 5.477$ $sqrt(30) = +-5.477$

And therefore:

$\pm 3 \sqrt{30} = \pm 3 \cdot \pm 5.477 = \pm 16.432$ $+-3sqrt(30) = +-3 *+- 5.477 = +-16.432$ rounded to the nearest thousandth.

Answer 2 · 2017-04-30T00:03:55

A = $3 \cdot \sqrt{30}$ $3 * sqrt30$

Explanation:

Find a $\sqrt{}$ $sqrt$ which can be divided by the whole number:
= $\sqrt{30}$ $sqrt30$

Then find the quotient of the number and the divisor ( $\sqrt{}$ $sqrt$ ):
= 270 / $\sqrt{30}$ $sqrt30$
= 3
= 3 * $\sqrt{30}$ $sqrt30$

Answer 3 · 2017-04-30T00:23:25

$\sqrt{270} = 3 \sqrt{30} \approx \frac{15873}{966} \approx 16.431677$ $sqrt(270) = 3sqrt(30) ~~ 15873/966 ~~ 16.431677$

Explanation:

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

Next note that if $a, b \geq 0$ $a, b >= 0$ then:

$\sqrt{a b} = \sqrt{a} \sqrt{b}$ $sqrt(ab) = sqrt(a)sqrt(b)$

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

Also, if $a \geq 0$ $a >= 0$ then:

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

So we find:

$\sqrt{270} = \sqrt{9 \cdot 30} = \sqrt{9} \sqrt{30} = 3 \sqrt{30}$ $sqrt(270) = sqrt(9*30) = sqrt(9)sqrt(30) = 3sqrt(30)$

This is the simplest form of the principal square root.

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

We can calculate approximations to $\sqrt{30}$ $sqrt(30)$ using its continued fraction. Note that $30 = 5 \cdot 6$ $30=5*6$ is of the form $n (n + 1)$ $n(n+1)$ . 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:

$sqrt(30) ~~ 5+1/2 = 11/2$

$sqrt(30) ~~ 5+1/(2+1/(10+1/2)) = 241/44$

$sqrt(30) ~~ 5+1/(2+1/(10+1/(2+1/(10+1/2)))) = 5291/966$

Let's stop there and use this to give us an approximation for $sqrt(270)$ ...

$sqrt(270) = 3sqrt(30) ~~ 3*5291/966 = 15873/966 ~~ 16.431677$

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How do you find the square root of 270?

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