Question

What is the product of \[3\sqrt 4 \] and \[4\sqrt 8 \].

Answer 1

Hint: Here the given question based on the Multiplication and Division of Radicals, we have to multiply the given radicals. First, we should write the radical in exponent form like \[\sqrt x = {x^{\dfrac{1}{2}}}\] after to multiply use the one of the rule of exponent i.e., \[\sqrt a \times \sqrt b = \sqrt {ab} \] and on further simplification 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
\[ \Rightarrow \,\,\,3\sqrt 4 \times 4\sqrt 8 \]
Now multiply the radicands using the power of exponent \[\sqrt a \times \sqrt b = \sqrt {ab} \] and multiply their coefficients separately.
Where a=4, b=8 and 3, 4 are coefficients of radicands, then
\[ \Rightarrow \,\,\,3 \times 4\sqrt {4 \times 8} \]
On multiplying, the number inside the radicand and their coefficients separately
\[ \Rightarrow \,\,\,12\sqrt {32} \]
We can write 32 as \[32 = 16 \times 2\], the n
\[ \Rightarrow \,\,\,12\sqrt {16 \times 2} \]
As we know, 16 is the square number of 4, then
\[ \Rightarrow \,\,\,12\sqrt {{4^2} \times 2} \]
\[ \Rightarrow \,\,\,12\sqrt {{4^2}} \times \sqrt 2 \]
\[ \Rightarrow \,\,\,12 \times 4 \times \sqrt 2 \]
\[ \Rightarrow \,\,\,48\sqrt 2 \]
Or by using calculator
\[ \Rightarrow \,\,\,48\sqrt 2 \simeq 67.882251\]
Hence, it’s a required solution.

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.