Question

How do you simplify: square root of 32 \[ - \,\]square root of 18 ?

Answer 1

Hint: Here the given question based on the subtraction of Radicals, we have to subtract the given radicals. First, we should convert the two radicals as radicals by using the square factors of the number and next by subtracting them like radicals we get the required solution.

Complete step by step solution:
The square root of a natural number is a value, which can be written in the form of \[y = \sqrt a \]. It means ‘y’ is equal to the square root of a, where ‘a’ is any natural number. We can also express it as \[{y^2} = a\].Thus, it is concluded here that square root is a value which when multiplied by itself gives the original number, i.e., \[a = y \times y\].
The symbol or sign to represent a square root is ‘\[\sqrt {} \]’. This symbol is also called a radical. Also, the number under the root is called a radicand.
Consider the given expression:
Square root of 32 = \[\sqrt {32} \]
Square root of 18 = \[\sqrt {18} \]
Let subtract the two radicals:
\[ \Rightarrow \,\,\,\sqrt {32} - \sqrt {18} \]
Now multiply the radicands using the power of exponent \[\sqrt a \times \sqrt b = \sqrt {ab} \]
Where: \[\sqrt {32} = \sqrt {16 \times 2} \] and \[\sqrt {18} = \sqrt {9 \times 2} \], then
\[ \Rightarrow \,\,\,\sqrt {16 \times 2} - \sqrt {9 \times 2} \]
Or it can be written as:
\[ \Rightarrow \,\,\,\sqrt {16} \sqrt 2 - \sqrt 9 \sqrt 2 \]
As we know the 16 is the square number of 4 i.e., \[\sqrt {16} = \sqrt {{4^2}} = 4\]
And 9 is the square number of 3 i.e., \[\sqrt 9 = \sqrt {{3^2}} = 3\], then
\[ \Rightarrow \,\,4\sqrt 2 - 3\sqrt 2 \]
On subtracting, we get
\[ \Rightarrow \,\,\sqrt 2 \]

Hence, the required solution is \[\sqrt 2 \].

Note: The exponential number is defined as the number of times the number is multiplied by itself. It is represented as \[{a^n}\], where a is the numeral and n represents the number of times the number is multiplied. For the exponential numbers we have a law of indices and by applying it we can solve the given number. Remember when doing the addition and subtraction of exponential and radicand we have to make the like terms first.