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# How do you find the exact square root of 39?

Last updated date: 25th Jul 2024
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Hint: To find the exact square root of 39, we are going to use the method of log. First of all, let the square root of 39 be x. Now, square root can also be denoted by raising the number by $\dfrac{1}{2}$ .So, we will get the equation as $x = {\left( {39} \right)^{\dfrac{1}{2}}}$ .Now, take log on both sides and then simplify RHS. After that, to find the value of x, take antilog on both sides and we will get our answer.

In this question, we are supposed to find the exact square root of 39.
We can easily find this using a calculator, but we are going to see a method to find the square root of any number without using a calculator.
For this, we are going to use the log method.
First of all, the square root of a number means the number which when multiplied two times will give the original number. Square root of a number is denoted by $\sqrt {}$ .
Let the square root of 39 be $x$ .
$\Rightarrow x = \sqrt {39}$
We can also write square roots as raised to $\dfrac{1}{2}$ .
$\Rightarrow x = {\left( {39} \right)^{\dfrac{1}{2}}}$ - - - - - - (1)
Now, to find the square root of a number using the log method, introduce log on both sides of the equation.
Therefore, equation (1) becomes
$\Rightarrow \log x = \log {\left( {39} \right)^{\dfrac{1}{2}}}$ - - - - - - - - (2)
Now, we have the property $\log {a^b} = b\log a$ . Therefore, equation (2) becomes
$\Rightarrow \log x = \dfrac{1}{2}\log \left( {39} \right)$ - - - - - - - (3)
Now, the value of $\log 39 = 1.591064607$ .Remember that we have to find the exact value so write all the digits you get. Therefore, equation (3) becomes
$\Rightarrow \log x = \dfrac{1}{2}\left( {1.591064607} \right)$
$\Rightarrow \log x = 0.7955323035$
Now, we need the value of x. So, take antilog on both sides, we get
$\Rightarrow x = anti\log \left( {0.7955323035} \right) \\ \Rightarrow x = 6.244997998 \;$
Hence, the exact square root of 39 is $6.244997998$ .
You can also cross verify the answer using a calculator.
So, the correct answer is “ $6.244997998$ ”.

Note: Square root of a number is written under the $\sqrt {}$ sign and this sign is called a radical sign. The value inside this sign is called radicand. Here the important part in this question is that as we are finding the exact square root of 39, we need to write each and every digit after the decimal point. You can either use a log method or calculator to find the exact square root of a number. There is no other method.