Question

How do you find the value of $3$ square root of $8$ ?

Answer 1

Hint:
For finding the value of $3\sqrt 8 $ we need to solve the number inside the radical sign first. The number square root of $8$ can be expressed as $2\sqrt 2 $. Now find an approximate value of root two and substitute the value in $3 \times 2\sqrt 2 $ . The value of this can be found by multiplying the numbers together.

Complete step by step solution:
Here in this problem, we need to find the value of the number $3$ square root of $8$ i.e. $3\sqrt 8 $ .
Before starting with the solution, we must understand the fundamentals of square roots. The given expression in the question $3\sqrt 8 $ is a product of a positive whole number $3$ and a square root with the radical sign.
As we know the number $8$ is not a perfect square. It can be represented as:
$ \Rightarrow 8 = 2 \times 2 \times 2$
So the given product can be written as:
$ \Rightarrow 3\sqrt 8 = 3 \times \sqrt {2 \times 2 \times 2} $
Now we can take out $2 \times 2$ from the radical sign and solve it further
$ \Rightarrow 3\sqrt 8 = 3 \times \sqrt {2 \times 2} \times \sqrt 2 = 3 \times 2 \times \sqrt 2 $
The above expression can be simplified by multiplying the numbers
$ \Rightarrow 3\sqrt 8 = 3 \times 2 \times \sqrt 2 = 6\sqrt 2 $
Therefore, we get the final simplified value as:
$ \Rightarrow 3\sqrt 8 = 6\sqrt 2 $
To solve it further, we just need to multiply the value of $\sqrt 2 $ by substituting it.
The number $\sqrt 2 $ is an irrational number and does not have an exact value. An irrational number is a number that cannot be expressed as a fraction $\dfrac{p}{q}$ for any integers $p$ and $q$ . Irrational numbers have decimal expansions that neither terminate nor become periodic. Every transcendental number is irrational.
We can use the value of root $2$ as $\sqrt 2 = 1.414$
On substituting this value, we can obtain the required value:
$ \Rightarrow 3\sqrt 8 = 6\sqrt 2 = 6 \times 1.414 = 8.484$

Therefore, we found the value of $3$ square root of $8$ or $3\sqrt 8 $ as $8.484$.

Note:
In this question, we were given an expression $3\sqrt 8 $ to find the value but an irrational number, i.e. $\sqrt 2 $ can’t be expressed as decimal or fraction for the exact value. We can only find an approximate value of square roots of prime numbers. For example, $\sqrt 2 = 1.414$ is the square root of $2$ up to three decimal places.