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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.