Question

How do you simplify the square root of 24 divided by 2?

Answer 1

Hint: Here, we need to simplify the square root of 24 divided by 2. First, we will rewrite the number 24 as the product of its prime factors. Then, we will simplify it using the product rule of square roots. Finally, we will remove the common factors from the numerator and the denominator to find the required value of the expression.

Formula Used:
We will use the following formulas:
1.Rule of exponents: If two or more numbers with same base and different exponents are multiplied, the product can be written as \[{a^b} \times {a^c} = {a^{b + c}}\].
2.The product rule of square roots states that \[\sqrt {A \times B} = \sqrt A \sqrt B \], where A and B are real numbers.

Complete step-by-step answer:
We need to simplify the value of the expression \[\dfrac{{\sqrt {24} }}{2}\].
First, we will rewrite 24 as the product of its prime factors.
Let us use the divisibility tests of 2, 3, 5, etc. to find the prime factors of 24.
First, we will check the divisibility by 2.
We know that a number is divisible by 2 if it is an even number.
This means that any number that has one of the digits 2, 4, 6, 8, or 0 in the unit’s place, is divisible by 2.
We can observe that the number 24 has 4 at the unit’s place.
Therefore, the number 24 is divisible by 2.
Dividing 24 by 2, we get the number 12.
We can observe that the number 12 has 2 at the unit’s place.
Therefore, the number 12 is divisible by 2.
Dividing 12 by 2, we get the number 6.
The number 6 is also divisible by 2.
Dividing 6 by 2, we get the number 3.
The number 3 is a prime number.
Therefore, the prime factors of 24 are 2, 2, 2, and 3.
Therefore, we can rewrite the number 24 as
\[24 = 2 \times 2 \times 2 \times 3\]
Therefore, using the rule of exponents \[{a^b} \times {a^c} = {a^{b + c}}\], we can rewrite 24 as
\[\begin{array}{l} \Rightarrow 24 = {2^1} \times {2^1} \times 2 \times 3\\ \Rightarrow 24 = {2^2} \times 6\end{array}\]
Substituting \[24 = {2^2} \times 6\] in the expression \[\dfrac{{\sqrt {24} }}{2}\], we get
\[\dfrac{{\sqrt {24} }}{2} = \dfrac{{\sqrt {{2^2} \times 6} }}{2}\]
The product rule of square roots states that \[\sqrt {A \times B} = \sqrt A \sqrt B \].

Applying the product rule of square roots in the expression, we get

\[ \Rightarrow \dfrac{{\sqrt {24} }}{2} = \dfrac{{\sqrt {{2^2}} \sqrt 6 }}{2}\]

Simplifying the radical in the expression, we get
\[ \Rightarrow \dfrac{{\sqrt {24} }}{2} = \dfrac{{2\sqrt 6 }}{2}\]
The numerator and denominator have the common factor 2.
Removing the common factor 2, we get
\[ \Rightarrow \dfrac{{\sqrt {24} }}{2} = \sqrt 6 \]
Therefore, we have simplified the expression \[\dfrac{{\sqrt {24} }}{2}\] as \[\sqrt 6 \].

Note: We have used the term ‘prime factor’ in the solution. A prime factor is a factor that is a prime number. Prime numbers are the numbers which have only two factors, 1 and the number itself. For example: 2 is the product of 2 and 1. Therefore, the factors of 2 are 2 and 1. Thus, 2 is a prime number. Square root of a number is defined as a factor which when multiplied to itself gives the original number.