Question

How do you simplify the square root of \[28\] times the square root of \[14\]?

Answer 1

Hint: In this question, we have to find out the required value from the given particulars.
We need to first find out the square root of \[28\]. Then we need to find out the square root of \[14\], then we will multiply the two values. After doing the multiplication we can find out the required solution.

Complete step by step solution:
We need to simplify the square root of \[28\] times the square root of \[14\].That means the multiplication of the square root of \[28\] and the square root of \[14\].
 First we need to find out the square root of \[28\].
For that, we need to write it as a product of prime factors. Thus, we get,
\[ \Rightarrow \sqrt {28} = \sqrt {7 \times 4} = \sqrt {7 \times 2 \times 2} \].
Now, we need to find out the square root of\[14\] .For doing that, we also need to write it as a product of prime factors.
\[ \Rightarrow \sqrt {14} = \sqrt {7 \times 2} \]
Here, the square root of \[28\] times the square root of \[14\]
\[ \Rightarrow \sqrt {28} \times \sqrt {14} \]
On simplify we get
\[ \Rightarrow \sqrt {7 \times 2 \times 2} \times \sqrt {7 \times 2} \]
On rewriting we get
\[ \Rightarrow \sqrt {7 \times 2 \times 2 \times 7 \times 2} \]
Since for any real numbers,\[\sqrt A .\sqrt B = \sqrt {A.B} \]
\[ \Rightarrow 7 \times 2\sqrt 2 \]
On multiply the term and we get
\[ \Rightarrow 14\sqrt 2 \]

Hence, simplifying the square root of \[28\] times the square root of \[14\] we get, \[14\sqrt 2 \].

Note: Factorization:
In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization.
Square root:
In mathematics, a square root of a number x is a number y such that, \[{y^2} = x\]. In other words, a number y whose square is x.
For example, \[4, - 4\] are square roots of \[16\], because \[{4^2} = {\left( { - 4} \right)^2} = 16\].