# How do you find the square root of 529?

Sep 8, 2015

Check for divisibility by a perfect square to simplify. You will find that ${23}^{2} = 529$

#### Explanation:

When we try to simplify a square root, we look for perfect square factors.

Do this by testing perfect squares until you get to a number whose square if greater that $529$ (for example ${25}^{2} = 625$)

So we test ${2}^{2} = 4$. Clearly $4$ is not a factor of $529$ (nor is any other even number.

Test ${3}^{2} = 9$ which is not a factor. Skip ${4}^{2}$ because it is even.

Obviously ${5}^{2} = 25$ is not a factor. Skip ${6}^{2}$.
Keep going . . .

${23}^{2} = 529$ so we're finished.

$\sqrt{529} = 23$

May 17, 2017

Use a mixture of methods to find $\sqrt{529} = 23$

#### Explanation:

There are quite a few different ways to find square roots.

Here's a bit of a mish-mash for this particular example...

Given $529$, first split off pairs of digits starting from the right hand side to get:

$5 | 29$

Next, note that $5$ lies between two square numbers ${2}^{2} = 4$ and ${3}^{2} = 9$

We can approximate where $\sqrt{5}$ lies between $2$ and $3$ by linearly interpolating. What's that? We approximate the part of the graph of ${x}^{2}$ between $x = 2$ and $x = 3$ using a straight line to get our approximation.

Since $5$ is $\frac{1}{5}$ of the way between ${2}^{2} = 4$ and ${3}^{2} = 5$, we can approximate $\sqrt{5}$ by $2 + \frac{1}{5} = 2.2$

Hence a good first approximation for $\sqrt{500}$ is $22$.

How about $529$?

We can use the Babylonian method to find a better approximation.

Given a positive number $n$ and an approximation ${a}_{i}$ to its square root, a better approximation is given by the formula:

${a}_{i + 1} = \frac{{a}_{i}^{2} + n}{2 {a}_{i}}$

So putting $n = 529$ and ${a}_{0} = 22$, we find:

${a}_{1} = \frac{{a}_{0}^{2} + n}{2 {a}_{0}} = \frac{{22}^{2} + 529}{2 \cdot 22} = \frac{484 + 529}{44} = \frac{1013}{44} = 23.02 \overline{27}$

That looks suspiciously close to $23$ so try:

${23}^{2} = 529$

Thus $\sqrt{529} = 23$

Jun 30, 2017

$\sqrt{529} = 23$

#### Explanation:

Do a really rough estimate first using squares of multiples of $10$

${20}^{2} = 400 \mathmr{and} {30}^{2} = 900$

$529$ lies between $400 \mathmr{and} 900$

so $\sqrt{529}$ will lie between $20 \mathmr{and} 30$

Now look at the last digit ...$9$
There are only two numbers whose squares end with a $9$,

$3 \mathmr{and} 7 \rightarrow \text{ } {3}^{2} = 9 \mathmr{and} {7}^{2} = 49$

So the possibilities are $23 \mathmr{and} 27$

However, ${25}^{2} = 625 \mathmr{and} 529$ is less than $625$

My first guess would therefore be $23$

Multiplying confirms that $23 \times 23 = 529$

Hence $\sqrt{529} = 23$