Question

How do you simplify \[2\sqrt {1000} \]?

Answer 1

Hint:To solve this question, we need to find the square root of 1000 first. For that we will factorize 1000 as a product of prime numbers. To find the square root, we will find the pairs of similar [prime numbers so that we can get those pairs out of the square root as a single number. And the single prime number will remain under the sign of the square root.

Complete step by step answer:
We are given \[2\sqrt {1000} \].First we will write 1000 as a product of prime numbers.We know that the smallest prime number is 2. As the unit digit of 1000 is 0, we can divide it by 2.
$\dfrac{{1000}}{2} = 500 \\
\Rightarrow 1000 = 2 \times 500 \\ $
Again, 500 is also divisible by 2.
$\dfrac{{500}}{2} = 250 \\
\Rightarrow 500 = 2 \times 250 \\
\Rightarrow 1000 = 2 \times 2 \times 250 \\ $
250 is also divisible by 2.
$\dfrac{{250}}{2} = 125 \\
\Rightarrow 250 = 2 \times 125 \\
\Rightarrow 1000 = 2 \times 2 \times 2 \times 125 \\ $
As the unit digit of 125 is not zero or an even number, it is not divisible by 2. Therefore, we will now check the divisibility of 3 for the number 125.The sum of all three digits of the given number is $1 + 2 + 5 = 8$ which cannot be divided by 3. Thus, 125 is not divisible by 3.Now we will check for divisibility by 5. As the unit digit is 5, it is divisible by 5.
\[\dfrac{{125}}{5} = 25 \\
\Rightarrow 125 = 5 \times 25 \\
\Rightarrow 1000 = 2 \times 2 \times 2 \times 5 \times 25 \\ \]
25 is also divisible by 5.
\[\dfrac{{25}}{5} = 5 \\
\Rightarrow 25 = 5 \times 5 \\
\Rightarrow 1000 = 2 \times 2 \times 2 \times 5 \times 5 \times 5 \\ \]
Now, we can write
\[2\sqrt {1000} \\
\Rightarrow 2\sqrt {2 \times 2 \times 2 \times 5 \times 5 \times 5} \\
\Rightarrow 2\sqrt {{2^2} \times {5^2} \times 2 \times 5} \\
\Rightarrow 2 \times 2 \times 5\sqrt {2 \times 5} \\
\therefore 20\sqrt {10} \\ \]
Thus, our final answer is $20\sqrt {10} $.

Note:We have applied the division method for finding the prime factors of the given number. The general steps for this method are: First step is, dividing the given number by the smallest prime number. In this case, the smallest prime number should divide the number exactly. Second step is, again dividing the quotient by the smallest prime number. Next step is, repeating the process, until the quotient becomes 1. Final step is to multiply all the prime factors.