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How do you multiply the square root of x times 4th root of x times 9th root of x?

Last updated date: 13th Jun 2024
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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 law of indices i.e., \[{a^m} \times {a^n} = {a^{m + n}}\] 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 data in given question
square root of x times i.e, \[\sqrt x \]----(1)
4th root of x times i.e., \[\sqrt[4]{x}\]------(2)
9th root of x times i.e., \[\sqrt[9]{x}\]------(3)
Now multiply the radicands (1), (2) and (3) together, then
\[ \Rightarrow \,\,\,\,\sqrt x \times \sqrt[4]{x} \times \sqrt[9]{x}\]
Write the radical in exponent form, then
\[ \Rightarrow \,\,\,\,{\left( x \right)^{\dfrac{1}{2}}} \times {\left( x \right)^{\dfrac{1}{4}}} \times {\left( x \right)^{\dfrac{1}{9}}}\]
By law of indices, to multiply expressions with the same base, copy the base and add the indices i.e., \[{a^m} \times {a^n} = {a^{m + n}}\], then
\[ \Rightarrow \,\,\,\,{x^{\dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{9}}}\]
Take 36 as lcm in the power
\[ \Rightarrow \,\,\,\,{x^{\dfrac{{18 + 9 + 4}}{{36}}}}\]
On simplification, we get
\[ \Rightarrow \,\,\,\,{x^{\dfrac{{31}}{{36}}}}\]
Write this in the radicand form
\[ \Rightarrow \,\,\,\,\sqrt[{36}]{{{x^{31}}}}\]
Hence, the value of \[\sqrt x \times \sqrt[4]{x} \times \sqrt[9]{x}\] is \[\sqrt[{36}]{{{x^{31}}}}\].

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.