# How do you find the square root of 500?

Mar 2, 2018

the answer is $10 \sqrt{5}$

#### Explanation:

$100 \cdot 5$ gives you $500$ now $10 \cdot 10$ gives you $100$ now you have two $10 s$ and you get $5 , 2$ for both $10 s$ and you have a pair of $5$ and a pair of $2$ put the $5$ earlier doesn't have a partner so you keep him inside b/c he doesn't have anyone

Mar 2, 2018

We get $22.360$ up to three decimal places.

#### Explanation:

As the given number is not a perfect square, the square root of this number will be a decimal number.
We must, therefore, state the accuracy to which the square root is to be calculated.

Suppose we need to calculate the $\sqrt{500}$ to an accuracy of three decimal places.

Step 1. Make pairs of digits starting from Units place.
we get $5 \overline{00}$.

Step 2. Consider the unpaired digit and calculate the nearest digit which is just lower than the square root of this digit. We consider $5.$ The nearest digit is $2$ as ${3}^{2} = 6 > 5$.

We get first digit of the answer. Workout the remainder which $= 1$. Bring down the next pair which is $\overline{00}$. Makes the dividend $= 100$. Also add the answer digit to the divisor. We have $2 + 2 = 4$

Step 3. Place highest digit in units place of divisor to make the divisor which will divide dividend $100$. We see that it can be divided with $2 \times 42$. $\therefore 3 \times 43 > 100$.
Complete the division. Nearest digit is $2$. Complete the division and obtain remainder. $= 100 - 84 = 16$. Also add the answer digit so obtained to divisor. We get $42 + 2 = 44$ for the next step.

Step 4. Next we have decimal in the original number. Place it after the quotient $22$ and bring a pair of zeros down. The dividend becomes $1600$

Continue steps till three places of decimal.

We see that in the Step 5. Number remaining is $304.$ Adding two zeros makes it $30400$ We are not able to divide the dividend because even with digit $1$ placed at units place of divisor the product of quotient digit and divisor is $>$ the dividend. We place a $0$ in the quotient place and add two zeros to the dividend

2|bar(5bar00.bar(00)\ bar(00)\ bar00\ bar00(22.360
$\underline{2 | 4} \setminus \downarrow$
$42 | 100$
$\underline{\setminus 2 | \text{ } 84} \downarrow$
$443 | 1600$
$\underline{\text{ } 3 | 1329} \downarrow$
$4466 | 27100$
$\underline{\text{ } 6 | 26796} \downarrow \downarrow$
$44820 | 3040000$
$\underline{\text{ "0" }}$